Title 40 CFR Part 191 Compliance Certification

Application for the

Waste Isolation Pilot Plant

Appendix SECOTP2D

United States Department of Energy Waste Isolation Pilot Plant

Carlsbad Area Office Carlsbad, New Mexico

WIPP PA User's Manual for SECOTP2D

SECOTP2D USER'S MANUAL

Version 1.30

Rambiz Salari Rebecca Blaine

May 8,1996

SECOTPZD . Version 1.30 User's Manual . Version 1.01

WPO # 36695 A~ri l 18 . 1996

Page 2

Table of Contents

1.0 INTRODUCTION ................................................................ 1.1 Software Identifier .............................................................. 1.2 Points of Contact ................................................................

1.2.1 Code Sponsor .............................................................. ........................................................... 1.2.2 Code Consultant

.................................................................. 2.0 FWNCTIONAL REQUIREMENTS 3

...................................... 3.0 REQUIRED USER TRAINING AND/OR BACKGROUND 4

............................................. 4.0 DESCRIPTION OF THE MODELS AND METHODS 4

.................................... 5.0 CAPABILITIES AND LIMITATIONS OF THE SOFTWARE 4

................................................ 6.0 USER INTERACTIONS WITH THE SOFTWARE 5

................................................................... 7.0 DESCRIPTION-OF INPUT FILES 5

.................................................................................... 8.0 ERROR MESSAGES 5

9.0 DESCRIPTION OF OUTPUT FTLES ................................................................ 6

10.0 REFERENCES ......................................................................................... 6

11.0 APPENDICES .......................................................................................... 7 ............. Appendix I: SECOTP2D User's Manual. by Kambiz Salari and Rebecca 13laine* 8

......................................................... Appendix 11: Sample DiagnosticslDebug File 9 Appendix III: Review Forms .......................................................................... 12

*The User's Manual in Appendix I has its own pagination .

SECOTPZD, Version 1.30 W O # 36695 User's Manual. Version 1.01 April 18, 1996

Page 3 - 1.0 INTRODUCTION

This document serves as a User's Manual for SECOTPZD, as used in the 1996 WIPP PA calculation. As such, it describes the code's purpose and function, the user's interaction with the code, and the models and methods employed by the code. An example output fde is included for the user's convenience.

1.1 Software Identifier Code Name: SECOTP2D (Sandia-ECOdvnamicflrausPort in 2 Dimensions)

Prefix: ST2D2 Version Number: 1.30, April 18, 1996 Platform: FORTRAN 77 for OpenVMS AXP, ver 6.1, on DEC Alpha

1.2 Points of Contact

1.2.1 Code Sponsor Rebecca L. Blaine Ecodynamics Research Associates P.O. Box 9229 Albuquerque, NM 87 1 19 Voice: (505) 843-7445 Fax: (505) 843-9641

-. 1.2.2 Code Consultant Kambiz Salari Ecodynamics Research Associates P.O. Box 9229 Albuquerque, NM 871 19 Voice: (505) 843-7445 Fax: (505) 843-9641

- 2.0 FUNCTIONAL REQUIREMENTS

R. 1 This code performs single component radionuclide transport in fractured or porous media

R.2 This code performs multiple component radionuclide transport in fractured or porous media.

R.3 This code models single porosiry transport in fractured or porous media

R.4 This code uses a dual porosity model for fractured media with advective and diffusive components in the fractures and only a diffusive component in the matrix material.

R.5 This code models a point source.

R.6 This code calculates the discharge across a user defined boundary.

SECOTPZD, Version 1.30 WPO # 36695 User's Manual, Version 1.01 April 18, 1996

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REQUIRED USER TRAINING AND/OR BACKGROUND -

3.0

Code user prerequisites are described in detail in Appendix I.

4.0 DESCRIPTION OF THE MODELS AND METHODS

Models and methods for SECOTP2D are described in detail in Appendix I.

5.0 CAPABILITIES AND LIMITATIONS OF THE SOFTWARE

Capabilities and limitations of SECOTP2.D include the following:

SECOTP2D can compute multiple component solute transport in fractured porous media using discrete-fracture and dual-porosity models.

SECOTPZD allows for radioactive decay and generation of daughter products.

The matrix block equation can model both a matrix material and a clay lining.

For the fracture-with-matrix block system, transport in the fracture is produced by the combined effect of advection and hydrodynamic dispersion, while transport in the matrix block is assumed to be dominated by molecular diffusion.

SECOTPZD is an implicit f ~ t e volume code that is second-order accurate in space and time. It uses operator splitting, an Approximate Factorization procedure, the delta formulation, and a finite volume staggered mesh.

SECOTPZD models the advective terms using either Total Variation D i h i n g (TVD) or central differencing. TVD eliminates the unwanted oscillations near sharp gradients while maintaining high accuracy in the solution.

SECOTPZD uses a generalized three-level scheme for time discretization.

An Approximate Factorization technique is used to alleviate the shortcomings (large memory requirements and CPU time) of banded LU solvers. However, while Approximate Factorization reduces the operation count and memory usage for inverting the coefficient matrix, it does introduce complications in applying the implicit boundary conditions.

Implicit coupling of the equations for the fracture and matrix block is used. This approach is more robust than explicit coupling.

SECOTPZD, Version 1.30 User's Manual. Version 1.01

WPO # 36695 April 18, 1996

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The spatial accuracy of the SECOTP2D code, with the TVD option cn, is less than second-order accurate, and the deviation from second-order accuracy will depend on how many sharp fronts exist in the computational plane.

Lack of flow field smoothness near boundaries can pose difficulties for the transport calculation. The SECOTPZD code, in most cases. can eliminate these difficulties bv automatically ass i -eg the boundary conditions using the flow field.

SECOTP2D provides discharge calculations on different predefined, closed boundaries.

These capabilities and limitations of the software are discussed in detail in Appendix I.

6.0 USER INTERACTIONS WITH THE SOFTWARE

User interactions with the software are discussed in detail in Appendix I.

7.0 DESCRIPTION OF INPUT FILES

Input files for SECOTP2D are discussed in detail in Appendix I.

8.0 ERROR MESSAGES .-

SECOTPZD uses the following lines of code to report error messages. Errors cause program execution to abort.

These write statements generate an error message indicating that the maximum number of iterations has been exceeded.

WRITE(6,*) 'SEC02D-TRANSPORT:' W R I T E ( 6 , * ) 'ERROR, Subroutine FRACTURE, maximum numjer of ' WRITE(6 , ' ) ' i t e r a t i o n exceeded. MAXITER =',MAXITER WRITE(6, ') ' I t is possible intra-t ime step i t e r a t i o n i s ' WRITE(6,*) 'd iverging. ' I* "p. +: ', ' '. -.

To recover from this error, rerun SECOTPZD with a greater number of time steps. \----- "

These write statements generate an error message indicating that the corner point of discharge surface is not in the computational domain.

WRITE(6, ' ) 'SEC02D-TRANSPORT: ERROR, Subroutine VEL.' WRITE(6 , ' ) 'Corner point of discharge surface is out o f ' ,

z a computational domain.'

0 These write statements generate an error message indicating that the value of the source function QC cannot be evaluated.

W R I T E ( 6 , * ) 'SECOZD-TRANSPORT: ' , > ' E r r o r , subroutine QC-INTEGRATE.'

W R I T E ( 6 , * ) 'value of source function QC cannot be'

SECOTPZD, Version 1.30 User's Manual, Version 1.01

Page 6 - > a evaluated for a specified time.'

WRITE(6,') 'TIME = ',TB

This error occurs when the time domain of the source function is not aligned with the time domain of the simulation. The error is corrected by ensuring that these time domains are aligned and then rerunning SECOTP2.D.

These write statements generate an ermr message indicating that location of a source point is not in the computational domain.

WRITE(6,*) 'SECOZD-TRANSPORT: ' , > 'ERROR, Subroutine SOURCE.'

WRITE(6,'l 'Location of a source point is out', > ' of computational domain.'

These write statements generate an error message indicating that an end-of-file has been detected.

WRITE(6,') 'SECOZD-TRANSPORT:' WRITE(^,*) 'ERROR, End-of-file detected (velocity)

9.0 DESCRIPTION OF OUTPUT FILES

Output files for SECOTP2D are discussed in detail in Appendix I. A sample diag~~osticsldebug file is provided in Appendix 11.

10.0 REFERENCES

References for SECOTP2.D are listed in Appendix I.

SECOTPZD. Version 1 .30 WF'O # 36695 User's Manual, Version 1.01 April 18. 1996

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11.0 APPENDICES

The following section provides the appendices for this document.

ABSTRACT

This is a user's manual for the SECOTP2D (Sandia-ECOdynamics/2-D'l1lle11~ion) code, Version 1.30. This program currently runs on an Alpha AXP computer with OpenVMS 6.1 operating system. SECOTP2D computes multiple component solute transport in fractured porous media using discrete-hcture and dual-porosity models. It allows for radioactive decay and generation of daughter products. In addition, the matrix block equation can model both a matrix material and a clay lining. For the hcture-with-matrix block systeq transport in the ffacture is pro.duced by the combined effect of advection and hydrodynamic dispersion, while transport in the matrix block is assumed to be dominated by molecular diffusion. SECOTP2D is an implicit finite volume code that is second-order accurate in space and time. It uses operator splitting, an Approximate Factorization procedure, the delta formulation, and a finite volume staggered mesh. The advective terms are modeled using either Total Variation Diminishing (TVD) or central differencing. T i e discretization is a generalized three-level scheme. The flow field for transport is obtained fkom the SECOFL2D code or equivalent. This manual presents the theory, numerical algorithm, and verification of the SECOTP2D code. Also, the p r e and postprocessors are described within the Compliance Assessment Methodology CONtroller (CAMCON) environment.

WIPP PA

User's Manual

for

SECOTP2D, Version 1.30

Document Version 1.01

WPO # 36695

April 18, 1996

CONTENTS

1 INTRODUCTION .................................................................................................................. 1 . . ..................................................................................................... 1.1 Code User Prerequmtes 1

2 GOVERNING EQUATIONS ................................................................................................ 2

3 NTJMERICAL DISCRETIZATION ................................................................................... 4 . . 3.1 Fmte Volume Grid ............................................................................................................ 4 3.2 Coordinate Transformation ............................................................................................... 4

........................................................................................ 3.3 invariants of the Transformation 4 .............................................................................................................. 3.4 Fracture Equation 4

3.5 Matrix Block Equation ..................................................................................................... 10 3.6 Fracture-Matrix Coupling ................................................................................................. 11 . .

3.6.1 Imphcit Coupling ....................................................................................................... 11 3.7 Solution Accuracy ............................................................................................................ 11

4 CODE VERIFICATION ..................................................................................................... 13 .................................................................................................. 4.1 Error Estimator and GCI 13

............................................................................................................ 4.2 Fracture Equation 14 ........................................................................................... 4.2.1 Steady State Calculations 15

. 4.2.2 Unsteady Calculations ................................................................................................ 18 . . ..................................................................................... 4.2.3 Implicit Boundary Conhbons 19 ..................................................................................................... 4.3 Matrix Block Equation 20

4.4 Coupling Procedure ......................................................................................................... 21 4.5 Multiple Species ............................................................................................................... 23

..................................................................................................... 4.6 Discharge Calculations 23

5 CODE CONTIRMATION PROBLEMS ............................................................ ........... 25 5.1 Convergence Demonstration on WIPP PA Problems .................................................. 2 5

..................................................................................................... 5.1.1 Fracture Transport 25 5.1.2 Dual-Porosity Transport ............................................................................................. 25

5.2 Dual-Porosity Solution of Sudicky and Frind .................................................................... 26 ............................................................................. 5.3 High Peclet Transport Case with TVD 27

.............................................................................................. 6 HARDWARE PLATFORMS 28

7 USER INTERACTIONS. INPUT AND OUTPUT FILES ............................................ 29 7.1 User interactions with PRESECOTP2D ........................................................................... 29

................................................................... 7.1.1 Exercising PRESECOTP2D Interactively 30 7.1.2 Exercising PRESECOTP2D from a Command Line ................................................... 31 . .

7.2 Descnpaon Of Input Fies ................................................................................................ 31 ................................................................................ 7.2.1 Input CAMDAT Database Fie 3 1

.................................................................... 7.2.2 SECOFL2.D Transfer Velocity Data Fie 32 ......... 7.2.3 Input Source CAMDAT Database ............................................ .................... 32

Y. 1 * \'.

7.2.4 Input Control Fie ....................................................................................................... 32 - ......................................................................................... 7.3 Input Fie for PRESECOTP2D 49

7.4 Input Parameter Checking ................................................................................................ 51 7.5 User Interactions With SECOTP2D ................................................................................. 52 . . ................................................................................................ 7.6 Descnphon Of Input Files 53

7.6.1 Input Control Fie ....................................................................................................... 53 7.6.2 Property Data Input and Velocity Data Input Files ................................................... 53 . . .............................................................................................. 7.7 Descnpaon Of Output Fies 53 7.7.1 Binary Output Fie ...................................................................................................... 53 7.7.2 DiagnosticdDebug Output Fie ................................................................................... 53

.................................................................................... 7.8 Postprocessor, POSTSECOTPZD 53

8 TEST PROBLEM ................................................................................................................ 55 8.1 PRESECOTP2D ............................................................................................................. 55 8.2 SECOTP2D ........................................................ ........................................................... 57 8.3 POSTSECOTP2D .................................................................................. , ......................... 57

REFERENCES ........................................................................................................................ 58

LIST OF FIGURES

Figure 1 Schematic of dual-porosity model. ............................................................................. 66 Figure 2 Schematic of finite volume staggered mesh showing internal and ghost cells.

The concentrations are defined at cell centers and velocities at cell faces. ................ 67 Figure 3 Fracture-Matrix coupling verijication: comparison of computed hc ture solution

to the Tang analytical solution ................................................................................. 68 Figure 4 Fracture-Matrix coupling verification: comparison of computed matrix solution

to the Tang analytical solution ................................................................................. 69 Figure 5 Muitiple species verification: comparison of computed hc ture solution to the

Lester et. al. analytical solution for species 1. .......................................................... 70 Figure 6 Multiple species verification: comparison of computed fracture solution to the

Lester et. al. analytical solution for species 2 and 3 .................................................. 71 Figure 7 Convergence test on PA problem, vector 2, temporal behavior of the source

function. ................................................................................................................. 72 Figure 8 Convergence test on PA problem vector 2, fiacture transport, concentrations

contours and breakthrough curves for coarse grid 46x53 ......................................... 73 Figure 9 Convergence test-on PA problem, vector 2, hcture transport, concentrations

contours and breakthrough curves for medium grid 93x107. ................................... 74 Figure 10 Convergence test on PA problem, vector 2, hcture transport, concentrations

contours and breakthrough curves for fine grid 187x215. ........................................ 75 - Figure 11 Convergence test on PA problem, vector 52, dual-porosity transport, temporal

behavior of the source function. .............................................................................. 76 Figure 12 Convergence test on PA problem, vector 52, dual-porosity transpoq

concentrations contours and breakthrough curves for coarse grid 46x53. ................ 77 Figure 13 Convergence test on PA problem, vector 52, dual-porosity transport,

concentrations contours and breakthrough curves for medium grid 93x107. ............ 78 Figure 14 Convergence test on PA problem, vector 52, dual-porosity transpoq

.................. concentrations contours and breakthrough curves for fine grid 187x215 79 Figure 15 Example problem: Sudicky-Frind, time = 500 days, comparison of grid

convergence study to the analytical solution in the matrix. ....................................... 80 Figure 16 Example problem: Sudicky-Frind, time = 500 days, comparison of computed

concernration profle to the analytical solution in the h a e . ................................. 81 Figure 17 Example problem: Sudicky-Frind, time = 500 days, comparison of computed

concentration profile at different fracture locations to the analytical solutions in the matrix ........................................................................................................... 82

Figure 18 Example problem: Sudicky-Frind, steady state solution, comparison of com~uted concentration ~roiile to the analytical solution in the fracture. ................. 83 -

Figure 19 Schematic diagram of 2-D plane dispersion problem. ................................................ 84 Figure 20 Example problem: hcture transport for high Peclet case, analytical solution,

Pe=2.0 .................................................................................................................. 85 Figure 21 Example problem: hcture transport for high Peclet case, TVD Iimiter = 7; van

............................. .............................................. Leer's MUSCL limiter, Pe = 2.0. , 86 Figure 22 Example problem: fiacture transport for high Peclet case, TVD limiter = 1;

............................................................................. Upstream differencing, Pe = 2.0. 87

Figure 23 Example problem: fracture transport for high Peclet w e , analytical solution, - Pe = 10.0. ............................................................................................................... 88

Figure 24 Example problem: fracture transport for high Peck case, TVD limiter = 7; van . . .......................................................................... Leer's MUSCL h t e r , Pe = 10.0. 89 Figure 25 Example problem: hcture transport for high Peclet case, TVD limiter = 1;

........................................................................... Upstream differencing, Pe = 10.0. 90 Figure 26 Test problem for SECOTP2D using CAMCON environment, concentration

. LIST OF TABLES

Table 1: Partial list of schemes available ...................................................................................... 6 Table 2: Benchmark (I): steady state calculations. TVD limiter = 1; central differencing,

uniform grid ................................................................................................................ 15 Table 3: Benchmark (I): steady state calculations. TVD limiter = 3; minmod (1.r). uniform

grid ............................................................................................................................. 16 Table 4: Benchmark (I): steady state calculations. TVD limiter = 4; rninmod (1.2). uniform

grid ............................................................................................................................. 16 Table 5: Benchmark (I): steady state calculations. TVD limiter = 5; minmod (2.r). uniform

grid ............................................................................................................................. 16 Table 6: Benchmark 0: steady state calculations. TVD limiter = 6; minmod (2.2r). uniform . .

grid ............................................................................................................................. 16 Table 7: Benchmark (I): steady state calculations. TVD limiter = 7; van Leer's MLTSCL.

uniform grid ............................................................................................................... 17 Table 8: Benchmark (I): steady state calculations. TVD limiter = 8; Roe's superbee. uniform

grid ............................................................................................................................. 17 Table 9: Benchmark 0: steady state calculations. TVD limiter = 1; central differencing, non-

uniform grid ................................................................................................................ 17 Table 10: Benchmark (I): timedependent calculations. trapezoidal time differencing. TVD

limiter = 1; central differencing, uniform grid ............................................................... 18 Table 11: Benchmark (I): time-dependent calculations. trapezoidal time differencing, TVD

limiter = 7; van Leer's MUSCL. uniform grid .............................................................. 18 Table 12: Benchmark (I): time-dependent calculations. trapezoidal time differencing, TVD

limiter = 1; central differencing, non-uniform grid ...................................................... 19 Table 13: Benchmark (I): time-dependent calculations. 3-point backward time differencing,

TVD limiter = 1; central differencing, non-uniform grid .......................................... 19 Table 14: Benchmark 0. time-dependent calcuiations. 3-point backward time differencing,

...................................................... TVD limiter = 1; central differencing, uniform grid 20 Table 15: Matrix vedication: steady state calculations. uniform grid ......................................... 21 Table 16: Matrix verification: steady state calculations. non-uniform grid .................................. 21 Table 17: Matrix verification: unsteady calculations. uniform grid ............................................. 21 Table 18: Half-lives and equilibrium constants of nuclides .......................................................... 23 . Table 19: Discharge calculations. uniform grid, benchmark (I) ................................................... 24 Table 20: Discharge calculations. stretched grid, benchmark (I) ................................................. 24 Table 21: Hardware. operating systems. and machine-zeros ....................................................... 28 Table 22: Computed results on different platforms benchmark (I) .............................................. 28

SECOTP7.D. Version 1.30 User's Manual, Version 1.01

WPO # 36695 April 18. 1996

.-. Page 8

Appendix I: SECOTP2D User's Manual, by Kambiz Salari and Rebecca Blaine

1 INTRODUCTION

The SECOTPZD code, Version 1.30, performs multiple component transport in eactured porous media. The media are represented using discrete-&awe and dual-porosity models. In the discrete-hcture model, the system is assumed to have multiple planar fractures (slab geometry). The goveming equations include the effect of advection, hydrodynamic dispersion, linear radioactive decay and generation, and an assumption of linear equilibrium isotherms. In the dual-porosity model, the system is comprised of numerous ffactures and a porous rock matrix. The matrix goveming equations represent the effect of diffusion, decay, and generation. This equation canmodel both the porous matrix material and a clay lining. The flow field, which can be either steady or unsteady, is obtained &om the SECOFL2D code [I] or equivalent.

SECOTP2D is an implicit finite volume code that is second-order accurate in space and time. It uses operator splitting, an Approximate Factorization procedure, the delta formulation, and a finite volume staggered mesh. The advective terms are modeled using either Total Variation Diminishing (TVD) or central differencing schemes. Time discretization is a generalized three- level scheme. SECOTP2D features manual and automatic boundary dehition which can vary ffom cell to cell. The code has modern FORTRAN structure and due to its operator splitting scheme is quite &cient 3 execution time and memory usage. The code has an option of computing the history of integrated discharge around any number of defined closed boundaries within the computational mesh. This manual describes the governing equations, the development of numerical schemes, code

verification and machine dependency, and detailed explanations of how to use Version 1.30 of the SECOTPZD the code @re- and postprocessors) in the environment of the Compliance Assessment Methodology Controller (CAMCON).

The SECOTP2D code is a part of SECO suite of codes that includes flow, trausport, and particle tracking in 2- and 3-dimensions. SECOTP2D has been referred to as SECO- TRANSPORT in the open literature [I, 2,3,4].

1.1 Code User Prerequisites

In order for the theoretical section of this manual to be useful the user will need the following: <.

>.#' , ~. . *... An understanding of partial dZerential equations. ? , ~ . . ".' 5

: , .; - I.

Senior level undergraduate course in linear algebra. . , , . , , ; ,I . .

Graduate or senior level undergraduate course in numerical methods. !: , . \ .. ... li I . . 'i' : (

Fist level undergraduate course in groundwater flow and transport. , + - . . . ' 1

In order to run the code and the associate pre- and postprocessors, an understanding of the CAMCOM environment is required. To apply SECOTP2D effectively, the user should be aware of the code's capabiities and limitations. It is recommended that the user run some of the problems provided to gain understanding in using the code.

2 GOVERNING EQUATIONS

The equation for the transport of the kth radionuclide component in a porous media (N species) can be written as (for more detailed derivation of this equation see references [S, 6, 7, 81)

where k = 1, ...$ and each dependent variable ' Ck @fX3) is the concentration of the kth radionuclide. Fork = 1, the term involving Ck.1 is omitted. Physical parameters include D (~'0, .

a 2 x 2 hydrodynamic dispersion tensor, V (UT), the specific discharge; 4 (I), the effective porosity defined as the ratio of fi-acture pore volume per unit volume of porous medium; & (I), the retardation coefficient; hk (IL), the species decay constant; ck (J4LL3), the concentration of the kth injected radionuclide; Q (14, the well specific injection rate defined as the voIume of liquid injected per unit volume of porous medium; and Tk (M/TL3), the mass transfer term between the fiacture and the matrix. The hydrodynamic dispersion tensor [6] is d&ed as

where aL and ar are the longitudinal and transverse dispersivities (L), D; is the fiee water molecular diffusion coefficient (~'m, r is the tortuosity (I), and u and v are the components of the Darcy velocity or specific discharge (Lq.

The N fi-acture equations are linear and sequentially-coupled. Three different boundary conditions, Dirichlet, Neumann, and flux, are available which can be used on different parts of the rectangular domain.

The flow field V is assumed to be. independent of the solute concentration. In practice, the flow-field is obtained fiom the SECOFL2D code [I].

Since the dualantinuum model [6,9] includes the exchange of mass between the matrix block and the fracture, it is necessary to solve a a p o r t equation in the matrix block Assuming that there is no fluid flow (advection) in the matrix, the equation for the concentration of the kth species is given by [5].

where x is the coordinate originating &om the symmetry line of the matrix block, the prime denotes matrix properties, D' =Dl< is the coefficient of the molecular diffusion where r' is the matrix tortuosity. The remaining symbols have the same meaning as those in Eq. 1. AU variables in Eq. 3 are spatially dependent which gives the needed flexibiity to model a clay lining. - -

The equations for the hc ture and the matrix block are coupled through a mass transfer term I;, which, for a slab block model (Figure I), is given by

where 2 is a coordinate system originating from the center (symmetry line) of the matrix block, b is the fracture aperture, b' is the matrix block length, and 4 is the kacture porosity defined as kacture volume per unit volume of porous medium.

For a typical matrix slab, the initial and boundary conditions are given by

where 5 is a parameter characterizing the resistance of a thin skin adjacent to the hcture. This parameter is dehed as 5 = b a , where b, and D, are the skin thichess and the skin diffusion - coefficient, respectively.

3 NUMERICAL DISCRETIZATION

3.1 Finite VoIume Grid

SECOTPZD uses a finite volume staggered mesh as shown in Figure 2. The velocity components (u,v) are defined at cell faces and concentrations are evaluated at cell centers. The boundary conditions are applied using ghost cells. These additional cells are necessary to construct boundary conditions since there is no concentration information on cell boundaries.

3.2 Coordinate Transformation

The governing equations are transformed *om Cartesian coordinates (physical space) to stretched Cartesian coordinates (computational space). The transformations are chosen so that the grid spacing in the computa&onai space is &for& and of unit length. This produces a computati& space which is a rectangular domain with a uniform mesh. The standard unweighted differencing scheme can now be applied in the numerical formulation. The metrics of the transformation are space dependent. To transform the governing equation fiom physical to computational space, chain rule expansions Bre used to represent the Cartesian derivatives in the computational space. The transformed equations, through some algebraic manipulation, are written in strong conservation form. For additional idormation on coordinate transformatioq see Ref. [lo].

3.3 Invariants of the Transformation

In the process of t rdorming the governing equations, additional terms containing the derivatives of the transformation are generated. These terms are collected at the end of the transfonnation and are hown as the invariant of transformation. With the use of metric definitions, it can be shown that the invariants are analytically zero. There is an important problem associated with these invariants when equations are evaluated for uniform properties. If we require that the transformed governing equations satisfy the uniform conditions, the discretized form of the invariants must sum to zero. In 2-D, standard central differencing does satisfy this requirement.

3.4 Fracture Equation

Eq. 1 has been transformed into computational space using

where transformation metrics are 5, = Jv, % = Jib and J = 5,flr The transformed equation, with further algebraic manipulations, was put into a strong conservation form. This is done to ensure mass c o ~ o q which js essential here. The transformed equation is given by

where

Eq. 10 is solved using an implicit Approximate Factorization procedure [lo]. The advective terms are modeled by TVD [ll] and the remaining terms by central differencing. A generatized three-level implicit finite volume scheme, in delta form [lo], can be written as

where

Table 1: Partial list of schemes available 8 cp Schemes Truncation error 1 0 Euler, implicit 0 (At) % 0 Trapezoidal, implicit 0 1 3-point-backward, implicit 0 (AP)

The AC: be thought of as a correction to advance the solution to a new time-level (n+l). Eq. 20, with appropriate choice of the parameters 0 and cp, produces many two and three-level impticit schemes as shown in Table 1. Applying Eq. 20 to Eq. 10, we have

= RHS

The cross derivative terms are time-lagged (n-I) to facilitate the factorization of the right-hand side operator. The error introduced by lagging these terms a n be corrected through an intra-time step iteration.

The advective terms are modeled using the following TVD flux which we have developed [3] for staggered meshes. The flux is a combination of centered and locally upstream (or "upwiudd) weighted schemes. The basic concept of flux limiters (TVD and other algorithms) is to apply the upstream weighted scheme only locally and with only enough weighting to eliminate the non- physical oscillations which would be caused by centered differences. As the mesh is refined, the upstream weighting decreases, and the method is formally second-order accurate.

Physical space

where

Computational pace

where a = EJI and o = sign (a).

The function @ is called a limiter function [ I l l . SECOTP2D uses nine different limiter functions, six of which are TVD. The TVD limiters range from very compressive (Roe superbee) to very dissipative (minmod) [I I]. The limiter functions are defined as:

upstream differencing

central differencing

2. @(r)=w weighted upstream and central differencing

'a, 3. @(r) = minmod (5 r)

4. @(r) = minmod (5 2 r)

5. @(r) = minmod (2, r)

6. @(r) = minmod (2, 2 r)

8. @(r) = m m (0, min(2r, I), min (r, 2))

where r is defined as

van Leer's MUSCL

Roe superbee

o = sign (acn) and a is a wave speed The rninmod function is given by

minmod(4 b) = sign(a) mat (0, min (I a 1 , sign(a). b)) (26)

The SECOTP2D d e d t limitei is van Leer's MUSCL. The right hand side (RHS) of Eq. 22 involves terms that are at time level (n, -1 ) . After

evaluation of these terms, the solution at the new time level (MI) is obtained by

( I + ~ ~ L , +a,L, +s)A?= RHS

where I is an identity matrix, Lg and L, are the x and y operators, and S is a source term. To solve Eq. 27 for A', a two-dimensional banded matrix inversion is required. The banded LU solvers are adequate for small size problems; however, for large problems they require large amount of memory and CPU time. To deviate these shortcomings, the two-dimensional operator can be Approximately Factored to represent two one-dimensional operators. This decreases the operation count for the inversion of the coefficient matrix and reduces the memory usage. The implementation of implicit boundary conditions are quite different for the factored and unfactored operators. In the case of a I11 two-dimensional operator, the implicit boundary conditions are applied in a straight-forward manner with no complications; however, this is not the case for the factored operators in which intermediate boundary conditions are needed to construct the implicit boundary conditions. The Approximate Factorization technique has the advantage of reducing the operation count and the memory usage for inverting the coefficient matrix, but it introduces severe complications in applying the implicit boundary conditions. With one-dimensional operators, the solution at the new time level (MI) is obtained through the following sequence

(I +a,& + U , S ) A ~ = RHS

r?

At this point, Eq. 36 can be substituted into Eq. 31 and the boundary definition for the left boundary is complete. The right boundary for the r-sweep can be obtained in a similar manner.

For the y-sweep a general ghost cell boundary equation (bottom boundary) is written as

where d, e, and f are constants. This information is sufficient to update boundary stencils for this sweep.

Using the above procedure, the boundary information is added to the implicit part of each sweep (coefficient matrix) and then the inversion process is carried out. It should be noted that with this formulation there is no restriction on what type of boundary condition is used and the type may change &om cell to cell.

3.5 Matrix Block Equation

Using a similar procedure outlined for the fixture Eq. 1, the matrix block Eq. 3 is fist mapped to a computational space -

where

Then, Eq. 38 is discretized using the general implicit finite volume scheme, in a delta form given by Eq. 20.

where

Eq. 41 is solved using a tridiagonal inversion with implicit boundary conditions. The above procedure is second-order accurate in space and time.

3.6 Fracture-Matrix Coupling

The equations for the fracture and the matrix block are coupled through a mass transfer term Tk. This term is proportional to the gradient of the solute concentration in the matrix block at their interface. A simple approach to couple these equations is to time lag the r k term or, in other words, treat the coupling term explicitly. Our experience with expljcjt coupling has shown that if the molecular diffusion coefficient is high, and if there exists a clay lining, or there is a fine mesh resolution at the interface, the solution for the coupled system can go To make the coupling more robust, the equations must be coupled in a fully implicit manner. A procedure outlined in [a was adapted and redeveloped for an AF algorithm with the delta formulation and a finite volume grid. This new procedure, which is adopted here, couples the equations implicitly and has proved to be quite robust.

3.6.1 Implicit CoupIing

The coupling procedure consists of three steps. Step 1 involves writing the incremental mass transfer term Af; in the following form that can be inserted into the implicit part of the hc ture equation

Step 2 involves the evaluation of a and b terms. This is accomplished using the inversion process (LU factorization) in the solution of the matrix equation. After the construction of the lower tridiagonal matrix L and the intermediate solution, there is enough information to evaluate the A terms. This new information is fed into the fracture equation which subsequently is solved for concentrations in the eacture at the new time level (HI). Step 3 involves constructing the boundary condition for the matrix equation at the fracture-matrix interface using hcture concentrations at the ( H I ) time level. Matrix concentrations are then obtained using the upper

"-

tridiagonal matrix U by back substitution.

3.7 Solution Accuracy ;

The present code uses TVD limiters with three-level time differencing and directional ,/

improve accuracy and execution time. The code is second-order accurate both in time and space

- with appropriate time and spatial discretizations; however, TVD limiters in the vicinity of a sharp spatial gradient such as a propagating fiont make the solution locally first-order accurate. The

spatial accuracy of the SECOTP2D code with the TVD option on is less than second-order .-

accurate, and the deviation from second-order accuracy will depend on how many sharp fronts sdst in the computational plane. Problems with a moderately high Peclet number would greatly benefit from the TVD scheme by avoiding spurious oscillations commonly associated with the central differencing schemes. The long time scales of the problems to which the code is to be applied dictate the use of implicit algorithms.

The flow field is usually computed by SECOFL2D, but any comparable flow code can be used. It is important to note that the convergence tolerance on the flow must be smaller in magnitude than the source (Q) for the transport calculation. Lack of proper iterative convergence in the flow calculation can show up as a source term in the transport calculation due to its conservative formulation and in some cases can lead to instabilities.

In practice, the transport code is sensitive to a flow field which is erratic (non-smooth) at the boundaries. Any flow field that exhibits significant flow structure near a boundary suggests that the location of the boundaries are incorrect, i.e., the conceptual modeling is inadequate. We assume that if the solution on the boundaries is not known, then they should be located far enough away from any major structures in the flow field

Lack of smoothness of the flow field near boundaries can pose difficulties for the transport calculation. The SECOTP2D code, in most cases, can eliminate these diEculties by automatically assigning the boundaj conditions using the flow field. This is done by comparing the magnitude of the normal velocity component for each cell on four sides of the computational domain to a value E which is taken to be a 0.1 % of the m x h u m velocity magnitude. For cells in which the normal velocity components are below E, the boundary conditions are set to Dirichlet boundary condition with zero concentration The boundary conditions for the remaining cells are set by -~ exambing the flow field. If the normal velocity component is outward the boundary condition is set to a Neumann condition with zero gradient and if it is inward the boundary condition is set to a Dichlet condition with zero concentration.

4 CODE VERIFICATION

The code, which has been developed based on the scheme described in the algorithm section, is veritied for temporal and spatial accuracy against analytical solutions. The single and multiple species fracture transport, matrix diffusion, and coupling procedure parts of the algorithm are veritied individually.

The transport in the fracture for a single species contains advection, diffusion, and decay, therefore, the benchmark should exercise these components of the numerical algorithm. Benchmarks (I) and @) are given by Knupp (see Appendix). Benchmark (I) includes a time dependent spatially variable velocity field and a spatially variable dispersivity field; however, the boundary conditions are not time dependent. Benchmark (II) is used to verify the implicit boundary condition of the code within the operator splitting and the Approximate Factorization scheme.

The matrix equation is verified against an analytical solution which provides spatial variabiity in the properties with time dependent boundary conditions.

Having verified the fi-acture and the matrix solution for the transport of a single species individually, the only remaining part of the numerical solution is the coupling between the two equations. This implicit coupling procedure is verified using the Tang [13] analytical solution.

For multiple species transport, the code is benchmarked against the analytical solution formulated by Lester et. aL [I41 which provides the solution for three species in a chain.

SECOTP2D also provides discharge calculations on different pre-defined closed boundaries.

- S i the discharge calculations involve integration, this process needs to be verified. This is done using benchmark (I).

Uniform and stretched grids are used in computing benchmark cases where analytical solutions exist. Errors are estimated by comparing the computed solutions with the analytical solutions and using either an t , - or iniinity-norm. The accuracy or the order of convergence of these solutions are ascertained usiig Roache's Grid Convergence Index GCI [15].

Following this section on code verification. further c o ~ a t i o n is presented in Section 5 on the ~udicky-grind problem.

4.1 Error Estimator and GCI

The Grid Convergence Index GCI [15] provides an objective approach to the waluation of uncertainty in grid convergence studies. The GCI is constructed from a grid refinement error estimator wbich is derived from the theoxy of generalized Richardson extrapolation The GCI is a numerical error band equal to 3 times the error estimate. Grid doubling is not a requirement for GCL This flexibility is an important asset for problems in which grid doubling is not feasible. The GCI is d&ed for the 6ne grid ( i i a coarse-he pair) as

314 GCI (fine grid) = - rP - 1

- where E is defined as

f- and& are coarse and fine grid solutions, r > 1 is the grid refinement ratio, and p is the order of the method.

The theoretical or expected order p can be verified experimentally by examining the ratio of GCIs. For example, two GCIs can be computed with three grid solutions, fiom the fine grid to the intermediate grid (Gal& and fiom the intermediate to the coarse grid ( G G ) . The theory predicts that

If the results fiom a grid refinement (with constant r) approximately satisfjr this relation, it (a) verifies the order p, and @) indicates that the asymptotic range is achieved. For additional information on how GCI is computed, see Refs. [15, 16, 171.

4.2 Fracture Equation

The transport in the hcture contains advection, diffusioq and decay, therefore, the benchmark should exercise these components of the numerical algorithm. An analytical solution to unsteady hcture transport is given by Knupp (see Appendix) that includes a time dependent, spatially - variable velocity field, and a spatially variable dispersivity field. The boundary conditions are time independent and have zero values on all boundaries. The analytical solution assumes zero &ee- water molecular diffusion, no dual porosity term, and single-species decay. The benchmark tests are organized as follows.

1. Steady State Calculations

time-independent boundary conditions uniform and stretched grid central ditferencing TVD limiters

- minmod( 1, r) - minmod( 1,Zr) - minmod( 2, r) - minmod( 2,2r) - van Leer's MUSCL -Roe superbee

2. unsteady Calculations

3-point backward and trapezoidal time differencing time-independent boundary conditions

0 central differencing and van Leer MUSCL limiter uniform and stretched grid

3. Implicit Boundary Condition Verifications

timedependent boundary conditions central differencing uniformgrid

4.2.1 Steady State Calculations

For the steady state calculations, parameters are: 0 5 x I 1 m, 0 y 1 m, 4 = 1.0, R = 1.0, h =

O.Ol/day, 01 = 0.0, a2 = 0.0, u, = 0.02 dday, v. = 0.01 dday, ah = 10.0 m, and a=, = 1.0 m. For additional information and definition of variables, see Appendix. AU the steady state computations are converged to machine zero. There were five grid sizes used in the convergence test and they are all related by grid doubling (r = 2): 10x10,20x20,40x40,80x80, and 160x160.

Case 1.1 - Uniform grid and TVD limiter = 1, central rnerencing. Table 2 presents the results for the convergence. test. The ratios of GCIs in column 5 have remained relatively constant and equal to 4; hence, the solutions are second order accurate (r = 2, P = 4, sop = 2).

Table 2: Benchmark 0: steady state calculations, TVD limiter =: 1; central J

Grid 1, -norm Max error GCI GCI ratio ( 7 9

10x10 4.0507E-3 7.5701E-3 - ..

Case 13 - Uniform grid and TVI) limiter = 36, minmod limiters. Tables 3-6 present the results for the convergence tests. Column 5 presents the ratio of successive GCIs. Among the minmod limiter functions, limiter number 4 which is minmod(1, 2r), shown in Table 4, has exhibited a second-order accurate solution. The remaining minmod limiters have shown an order between fist and second-order accurate behavior which is consistent with the way TVD limiters are constructed. The last GCI ratio for the he s t grid pair in Tables 3, 5, and 6 show a drop below 4. This behavior is often associated with computer word-length limitations.

Table 3: Benchmark a: steady state calculations, TVD limiter = 3; minmod (l,r), uniform grid.

Grid Pa -norm Max. error GCI GCI ratio (P) 10x10 3.3663E-3 6.9064E-3 20x20 7.3017E-4 1.557OE-3 41.997 % 40x40 2.2055E-4 5.9687E-4 8.119 % 5.17 80x80 1.1293E-4 2.993 1E-4 1.715 % 4.73

160x160 6.277OE-5 1.5754E-4 0.799 % 2.15

Table 4: Benchmark (I): steady state calculations, TVD limiter = 4; minrnod (1,2r), uniform grid.

Grid Pa -norm Max. error GCI GCI ratio (P) 10~10 4.0259E-3 7.635 1E-3 20x20 9.7757E-4 1.9328E-3 210.077 % 40x40 2.4267E-4 4.841 1E-4 48.476 % 4.15 80x80 6.0588E-5 1.2079E-4 12.011 % 4.04

160x160 1.516OE-5 3.0198E-5 2.997 % 4.01

Table 5: Benchmark 0: steady state calculations, TVD limiter = 5; minmod (2,r), uniform grid.

Grid e, -norm Max error GCI GCI ratio (P) lOxl0 3.3666E-3 6.9072E-3 20x20 7.2917E-4 1.5517E-3 42.026 % 40x40 2.1956E-4 5.9444E-4 8.120 % 5.17 80x80 1.1279E-4 2.9866E-4 1.701 % 4.77

160x160 6.2757E-5 1.5745E-4 0.797 % 2.13

Table 6: Benchmark m: steadv state calculations. TVD limiter = 6: minmod (2,2r), unifork grid. '

Grid L, - nonn Max error GCI GCI ratio (P) lOxl0 5.3927E-3 9.7949E-3

Case 1.3 - Uniform grid and TVD limiter = 7, van Leer's MUSCL limiter. Table 7 presents the results for the convergence test. As the results in column 5 show, the solutions are second- order accurate. This Iimiter function is not too dissipative or compressive; because of this, it is chosen to be the default limiter for SECOTP2D under the TVD option.

Table 7: Benchmark m: steadv state calculations. TVD limiter = 7: van Leer's MLTSCL, uniform gih.

Grid P, -norm Max error GCI G€I ratio (9) 10x10 4.03 11E-3 7.6009E-3

Case 1.4 - Uniform grid and TVD limiter = 8, Roe's superbee limiter. Table 8 presents the results for the convergence test. This is a compressive limiter function and is designed to work with discontinuities in the solution. For this benchmark, which has a smooth solution, the limiter has shown second-order accurate behavior (column 5).

Table 8: Benchmark (I): steady state calculations, TVD limiter = 8; Roe's superbee, uniform grid.

Grid !, -norm Max error GCI GCI ratio (9) 7.7523E-3

Case 1.5 - Nonuniform grid and TVD limiter = 1, central differencing. Table 9 presents the results for the convergence test. The As corresponds to a minimum cell size. Geometric stretchings, with given initial cell sizes, are used to construct the grids. As the results in column 5 . .

show, the solutions are second-order accurate.

Table 9: Benchmark 0: steady state calculations, TVD limiter = 1; central differencing, non-uniform grid.

Grid A s 1, -nom Max. error GCI GCI ratio (9) 10x10 0.04 9.5779E-3 2.7666E-2 -,..-.

- , ". ',

20x20 0.02 2.2588E-3 6.0692E-3 217.352 % , . .. . ,..: . . . . 0.01 5.4968E-4 1.4608E-3 50.755 % 4.28 i ? . .

, . . 40x40 . ,

i s .i,, ,, ,* 80x80 0.005 1.3558E-4 3.5703E-4 12.297% 4.13 v;,. , j, i; . 160x160 0.0025 3.3674E-5 8.8283E-5 3.026 % 4.06 ,, .a ,"

4.2.2 Unsteady Calculations - For the time-dependent calculations parameters are: 4 = 1.0, R = 1.0, X = 0.001 l/day, rnl = 0.0157, oz= 0.00785, u.= 0.02 dday, v, = 0.01 &day, ah = 10.0 m, a~. = 10 m, and Time =

0.8 day. The numerical errors and the order of the schemes are evaluated in the same manner as was described previousiy.

Case 2.1 - Trapezoidal time diierencing (second-order), TVD limiter = 1, central differencing, and uniform grid. Table 10 presents the results for the convergence test.

Table 10: Benchmark (I): time-dependent calculations, trapezoidal time differencing, TVD limiter = 1; central diierencing, uniform grid.

Grid At t , -nom Max. error GCI GCI ratio (P) 10x10 0.04 3.3879E-3 6.8374E-3.

This exercise verifies the order of the overall scheme including the time dependent part of the algorithm. As the results in column 5 show, the solutions are second-order accurate in time and .

space. Case 2.2 - Trapezoidal time differencing (second-order), TVD limiter = 7, van Leer's

MUSCL limiter function, and uniform grid. Table 11 presents the results for the convergence test. Again, sohtions are second-order accurate as shown in column 5.

Table 11: Benchmark (I): time-dependent calculations, trapezoidal time differencing, TVD limiter = 7; van Leer's MSJSCL, uniform grid.

Grid At P, -norm Max. error GCI GCI ratio (9)

- Case 2.3 - Trapezoidal time diierencing (second-order), TVD limiter = I, central differencing, and non-uniform grid. As Table 12 shows, the results are second-order accurate.

Table 12: Benchmark 0: time-dependent calculations, trapezoidal time differencing, TVD limiter = 1; central differencing, non-uniform grid.

Grid As At e, -norm Max. error GCI GCI ratio (IP) 10x10 0.04 0.04 7.7854E-3 2.1 118E-2 20x20 0.02 0.02 1.891%-3 5.019OE-3 208.290 % 40x40 0.01 0.01 4.6400E-4 1.2186E-3 50.435 % 4.13 80x80 0.005 0.005 1.1448E-4 2.9891E-4 12.351 % 4.08

160x160 0.0025 0.0025 2.8298E-5 7.3579E-5 3.046 06 4.05

Case 2.4 - 3-point backward time differencing (second-order accurate), TVD limiter = 1, central differencing, and uniform grid. Table I3 presents the results for the convergence test. As the results show, solutions are second order accurate and are similar to rhat generated by the trapezoidal time differencing.

Table 13: Benchmark 0: time-dependent calculations, 3-point backward time diierencing, TVD limiter =i; central differencini, no&mi€orm grid.

Grid At P, -norm Max. error GCI GCI ratio (IP)

4.2.3 Implicit Boundary Conditions

Benchmark 0, which is given by Knupp (see Appendix), is used to verifl the implicit boundary conditions of the code. Since the computational domain is hite, the Dichlet boundary conditions are time dependent and may be obtained f?om the exact solution

Case 3.1: The parameters are: time = 25 days, u = 0.1 &day, a~ = 1.0 m, and as = 0.1 m. Four different grid sizes and time steps are used in this convergence study. The code was set to use 3-point backward time differencing which is second-order accurate and a TVD limiter = 1 (central differencing). Table 14 presents the computed solution to Benchmark 0, the error, and the GCIs. By examining the ratio of GCIs, it is evident that the overall solution is second-order accurate in time and space. Therefore, the implicit treatment of boundary conditions in the AF algorithm is verified to be second-order accurate in time.

Table 14: Benchmark 0, tirne-dependent calculations, 3-point backward time differencing, TVD limiter = 1; central differencing, uniform grid.

Grid Ax A t e, - nonn GCI GCI ratio (9) 20x20 0.05 0.25 7.697E-3

4.3 Matrix Block Equation

The numerical algorithm to solve the one-dimensional matrix block equation, described in Section 3.5, is verified against the following analytical solution. The governing equation is given by

and 0 I x I b. The variables D and p are defined as

The boundary and initial conditions are

The solution is

The convergence tests are done for steady state calculations using uniform and nonuniform grids' and for unsteady calculations with trapezoidal and 3-point backward time differencing with a uniform grid.

For steadv cases. the Dararneters are a = h = I and 6 = 4.0. All the steadv calculations are

.- As the behavior of the ratio of GCIs show (column 5), the solutions are second-order accurate. Table 16 shows the steady calculations with non-uniform grid. Again, from the r e d t s in column 5, the solutions are second-order accurate.

Table 15: Matrix vescation: steady state calculations, uniform grid. Grid P, -norm Max. error GCI GCI ratio (9)

*n-uniform grid. Grid As e, -norm Max. error GCI GCI ratio (f) 20 0.04 4.724OE-3 1.1464E-2 40 - 0.02 1.0251E-3 2.342s-3 244.200 % 80 0.01 2.465754 5.6313E-4 51.398 % 4.75 160 0.005 6.088s-5 1.3810E-4 12.259 % 4.19 320 0.0025 1.5147E-5 3.421 lE-5 3.020 % 4.06

For unsteady calculations, the parameters are a = 1, h. = 5, b = 4.0, and t = 1.0. Table 17 presents the convergence results for unsteady, trapezoidal time differencing and uniform grid. As the ratios of successive GCIs in column 5 show, the solutions are second-order accurate both in time and space. The results obtained with 3-point backward time differencing are similar to those computed by trapezoidal time differencing.

Table 17: Matrix verification: unsteady calculations, uniform gid. -- Grid A t P, -norm Max error GCI GCI ratio (f) 20 0.1 8.6435E-3 1.0795E-2 40 0.05 1.8384E-3 2.3088E-3 267.433 % 80 0.025 4.2851E-4 5.393 1E-4 55 407 % 4 83 160 0.0125 1.0353E-4 1.3041E-4 12.771 % 4.34 i

* - ,. 320 0.00625 2.544s-5 3.2062E-5 3.069 % 4.16

I

4.4 Coupling Procedure

To verify both the fracture and the matrix finite volume discretization as a system and the coupling procedure, we have chosen a dual porosity problem in one dimension with the analytical - solution given by Tang [13]. The Tang solution is numerically evaluated and hence there is an error associated with its evaluation. This is due to accuracy of the numexical algorithm used in

generating the solution In this case, the grid convergence test is not possible since the computed -- solution (uscact") is not accurate enough.

The fixture equation is

where 0 z < m. The fracture is along the z coordinate and the matrix block uses the x coordinate. The initial and boundary conditions are

The matrix equation is given by

where 6 5 x < m. The initial and boundary conditions are

For further explanation of the problem, definition of parameters, and the analytical solution, see - Re£ [13].

The test problem is set up by defining the required parameters as follows: the i k t u r e is along the x coordinate, fracture length, L = 10 m, hc tu re spacing 2.4 m Fracture properties: aperture, 6 = 1.0E-4 m, seepage velocity, V = 0.01 m/4 longitudinal dispersivity, = 0.50 m, moiecular diffusion coefticient, D = 1.382E-4 &Id, and hc tu re porosity, 4 = 0.42E-4. MatTtr properfies: the matrix uses the x coordinate (see Figure I), matrix porosity, = 0.01, and matrix diffusion coefticient, D' = 1.382E-7 m2/d RadionucIideproperties: decay constant, h = 0.154E- 3 114 and retardation factor, R = R' = 1. Inifid conditiom: c(x, 0) = cf(x, 0) = 0. The boundary conditions are

Fracture length and matrix block length are discretized using 80 and 15 stretched cells, respectively. The calculation was stopped at time = 100 days to test both spatial and temporal accuracy of the computed solution. Figures 3 and 4 present the comparison of the firacture and matrix solution to the analytical solution. The computed results reproduce the analytical solutions in both regions, which also verifies the accuracy of the coupling procedure. Further mesh refinement in both hc tu re and ma& block reproduced the same results.

4.5 Multiple Species

For multiple species transport, SECOTPZD is benchmarked against the Lester et. al. [14] analytical solution. This analytical solution, which includes the effect of axial dispersion, describes the transport of the radionuclide chains through a column of adsorbing media. Initially, alI species have zero concentrations and the solutions of Bateman equations are used as the time dependent inlet boundary conditions. For this exercise, we are using a three-member decay chain. The problem parameters are: time = 100 yr, length = 0.1 km, v = 0.1 Wyr, a~ = 0.5 km, and the species dependent properties are given in Table 18. The computed results were obtained with 80 cells in the column, At of 0.25 year, and 400 steps. Figures 5 and 6 present the computed and the analytical resuits for all the species. As these figures show, the computed results reproduce the analytical solutions.

Table 18: Half-lives and equilibrium constants of nuclides. Species Half-lives (yr) Decay Constant Vyr) Equilibrium Constant

1 15 0.04621 10,000

The ability of SECOTP2D to compute the discharge (mass flux) on a predefined closed boundary is v d e d using benchmark (I). The integral defining the discharge is given in the Appendix The numerical quadrature used is trapezoidal which is second-order accurate. Benchmark (I) is exercised in the same manner as desmied for case 1.1 (uniform grid) and case 1.5 (stretched grid). A single closed boundary which is a rectangular box is specified by using the coordinates of the upper left hand comer and the lower right hand comer of the box. For the uniform grid, the coordinates of these points are (0.2,O.X) and (0.8,0.2), and for the stretched grid they are (0.2085,0.7915) and (0.7915,0.2085). Table 19 and 20 present the results for the convergence study using five different grids. As the ratio of GCIs show, the computation of discharges are - second-order accurate for both uniform and stretched grid.

Table 19: Discharge calculations, uniform grid, benchmark (I) Grid Computed Error GCI GCI ratio (P)

10x10 0.411362 3.4856E-3

Table 20: Discharge calculations, stretched grid, benchmark (I) Grid As Computed Error GCI GCI ratio (P)

10x10 0.04 0.364443 3.182E-3 20x20 0.02 0.362069 8.089E-4 189.696% 40x40 0.01 0.361464 2.033E-4 48.409% 3.92 80x80 0.005 0.3613 11 5.085E-5 12.186 % 3.97

160x160 0.0025 0.361273 1.251E-5 3.065 % 3.98

5 CODE CONFIRMATION PROBLEMS

In the previous section 4, we have exercised a l l features of the code and have verified its accuracy in all aspects. The present section gives M e r confirmation calculations. Even if a code is rigorously verified, in the sense of a mathematical theorem, it can stiU be worthwhile to present selected additional confirmation calculations for the purpose of user confidence building.

5.1 Convergence Demonstration on WIPP PA Problems

To demonstrate convergence on typical problems, we will use two of the vectors fiom the 1992 WIPP PA calculation [IS].

5.1.1 Fracture Transport

Vector 2 F I E 2 scenario) was chosen for a grid convergence demonstration for fracture transport. This vector has moderate parameters, such as fracture aperture and hcture travel time.

Since we do not have an exact solution for vector 2, we rely on contours of the solution for judging convergence. We will use three different grid sizes, 46x53, 93x107, and 187x215. For each grid size three different time steps are used, At = 10, 5, and 2.5 years, to show time convergence. - Figure 7 shows temporal behavior of the source hc t ion over 10,000 years. Figures 8% 8c, and 8e present the contours of solute concentrations on the first grid at t = 10,000 years for three different time steps, respectively. The time resolution for this mesh is quite adequate since there is only slight cbange between contour plots. Figures Sb, 8 4 and 8f present breakthrough curves, with each plot presenting integrated discharges through three closed boundaries. The breakthrough curves also show convergence in time. Figures 9 and 10 show similar plots for grids number 2 and 3. As we r&e the grid, the plume becomes narrower and the concentration fiont becomes sharper. This is due to improved effectiveness of the TVD algorithm.

These sequences of grid and time steps clearly show that the problem is not adequately resolved on the coarse grid.

5.1.2 Dual-Porosity Transport

For a dual-porosity transport calculation, vector 52 (EIU scenario) is a typical example which has no extremes in its parameters. We will use the same grid sizes as in the fracture transport case, however, vector 52 has different time scales for both the fracture and the matrix block and requires different time steps, At = 2, 1, and 0.66 years.

Figure 11 shows temporal behavior of the source hc t i on over 10,000 years. Figures 12a, 12c, and 12e present the solute concentration on the fist grid at time = 10,000 years for the three different time steps, respectively. Similar to the fiacture calculation, the time resolution is satisfactory. Figures 12b, 12.4 and 12f present breakthrough curves. Again, there are no si@cant changes in the solution (visual detection) as the At decreases. Figures 13 and 14 show - a similar plot for grids 2 and 3. As we observed in the fracture calculation, the concentration

fiont becomes sharper as the grid becomes finer. Figure 12c shows some discharge on the side - boundary whereas on the h e r meshes there are no discharges. This points out that the first grid is not resolving the solution adequately; however, the other grids are adequate.

5.2 Dual-Porosity Solution of Sudicky and Frind

For dual-porosity calculations, SECOTP2D was benchmarked using Tang's [13] analytical solution. This solution has a limitation that the matrix block length is hirite. Sudicky and Frind published an analytical solution in 1982 [I91 for contaminant transport in a fractured porous media that removed this limitation. The original Sudicky-Frind analytical solution had some errors which were corrected by Davis and Johnston [20]. With this solution SECOTP2D can be exercised for a finite matrix block length in the dual porosity calculation. Van Gulick [21] has numerically implemented the Sudicky-Frind analytical solution. He has performed a thorough evaluation of the accuracy of the numerical solution and the range of parameters in which the solution converges. Herein, the data generated by van Gulick are used to compare with the unsteady and steady dual porosity calculations.

The following problem has been set up fiom the data provided by van Gulick The problem is a one-dimensional longitudinal transport in the fracture with transverse matrix diffusion. For the unsteady calculation, ihe solution is evaluated at 500 days.

Fracture properfies: aperture, b = 0.1E-3 (m); 4f = 1.9996E-4, longitudinal dispersivity, a~ =

0.1 (m); molecular diffusion coefiicient, Df = 1.3824E-4 (m2/day); pore velocity V = 0.0075 (m/day); decay constan!, X = 1.5366E-3 (Vday); fhcture retardation, R = 1.0; fracture length, L - = 2.0 (m). M& prop7ties: block length, 6' = 0.50 (m); matrix porosity, 4, = 0.01; diffusion

coefiicient, D'= 1.3824E-7 (m2/day); retardation, R= 1.0. The fhcture and the matrix concentrations are set to zero as the initial condition. The

boundary conditions for the fhcture are

where 01xsL. The i?acture was discretized with 200 uniform cells based on the grid convergence study for

the single porosity calculations. The matrix was discretized with 5 to 80 uniform cells. Figure 15 presents the grid convergence test for the matrix and the comparison to the analytical solution. The final solution was computed using 200 uniform cells in the fracture and 80 uniform cells in the matrix For the unsteady calculation, based on the convergence study, the time step was set to 1 day. The solution was converged to machine zero for steady state calculation.

Figure 16 presents the concentration profile in the hcture compared to the Sudicky-Frind analytical solution at 500 days; the agreement is excellent. Figure 17 shows the comparison of the concentration profiles in the matrix, at three different locations in the hcture, compared to the analytical solutions. Again, the agreement is excellent. Figure 18 presents the concentration -

profile in the fiacture at steady state compared to the analytical solution. Again, we have excellent agreement.

5.3 High Peclet Transport Case with TVD

To illustrate the advantages of the TVD algorithm presented in Chapter 3, we have chosen to solve a two-dimensional convection-dispersion problem for which we have an exact solution [22]. The medium is assumed to be homogeneous and isotropic with a unid'iectional steady-state flow. The initial solute concentration is zero. At t = 0 a strip-type source with a f%te length 2% as shown in Figure 19, along the y-axis is introduced. The source concentration remains constant with time. For additional information, see Ref. 1221.

The solution is obtained for two cases, low and high mesh Peclet numbers, where mesh Peclet number is defined as Ada& For both cases, a uniform 80x80 grid with 0 5 .r I 100 m and -50 s y s 50 m is used. The code was set up to use the TVD, van Leer's MLTSCL luniter, and trapezoidal time differencing.

Low mesh Peclet number care: Pe = 2, u = 1.0 dday, v = 0.0 dday, a = 0.625 m, a= =

0.0625 m, h = 0.0, a = 25 m, and time = 30 days. Figure 20 presents the analytical solution. Figure 21 presents fhe computed solution (Af = 0.1 day) using TVD, van Leer's MUSG limiter, and Figure 22 presents the computed solution using upstream differencing. It is clear fiom these figures that even at a low mesh Peclet number the upstream solution, due to its artificial diffusion, is not accurate; however, the TVD solution is very close to the analytical solution.

- High mesh Peclet number care: Pe = 10, u = 1.0 mfday, v = 0.0 m y , = 0.125 rn, ar = 0.0125 m, a = 25 m, and time = 30 days. Figure 23 shows results fiom the analytical solution. Figure 24 presents the TVD solution and Figure 25 presents the upstream solution (At = 0.1 day). The difference between these two solutions is dramatic. As expected, the TYD scheme rerained a sharp fiont; close to the analytical solution, as opposed to a very diffused front generated by the implicit upstream differencing.

As shown above, the TVD scheme in conjunction with second-order time discretization is more accurate in tracking sharp fkonts than the classical upstream schemes, even for low mesh Peclet number cases.

6 HARDWARE PLATFORMS

The portability of codes is always a question; therefore, it was essential to venfy that the computed results from SECOTPZD are not machine sensitive. The code was run, using benchmark (l) (case 1.1), on the following platforms: Digital Alpha, HP 9000/720, CRAY Y-MP C90, Silicon Graphics IRTS 4D-25, Sun 4c Spark station, and Micro-Vax IL The CRAY code is single precision while the others are double precision For simplicity, a coarse grid of 10x10 and 20 time steps were used. The -norm of the error, which is computed fYom the contributions of all the computational cells, and the maximurn error are good values to compare on different platforms. Initially, machine zeros were evaluated for all the computers. Table 21 presents the machine type, operating systems, and machine zeros. It is interesting to note that machine-zero were-the same for all computers (double precision) except for the CRAY (single precision) which was, about two orders of magnitude larger than the other machines. Table 22 presents the machine type, -norm of the error, and the maximum error (benchmark (I), case 1.1). As this table shows, all the computed results are the same within the prescribed precision of each machine. This clearly indicates that the results generated by the SECOTP2D code are not machine sensitive.

Table 21: Hardware, operating systems, and machine-zeros Hardware 0pemting System ~ a c h i n x e r o

Aloha OoenVMS 6.1 l.ll0E-16 Tie H P - ~ k ~ 0 9 . 0 1 , (unix) l.llOE-16

m y UNICOS 7.105E-15 IRIS IRIX 4.0.5, (unix) l.ll0E-16 S U ~ Sun 0s 4.1.1, (unix) l.ll0E-16

Micro-VAX II VMS V5.2 l.ll0E-16

Table 22: Computed results on different platforms benchmark (I) Hardware Max error

Alpha 3.910447413657691E-03 7.464205046390449E-03 lip 3.910447413657638E-03 7.464205046390337E-03

CRAY (SP) 3.9104474136439OOE-03 7.4642050463751OOE-03 IRIS 3.910447413657621E-03 7.464205046390448E-03 Sun 3.9104474136576OOE-03 7.4642050463903OOE-03

M~IXO-VAX 11 3.910447413657696''-03 7.464205046390449E-03

7 USER INTERACTIONS, INPUT AND OUTPUT FILES

In order to run SECOTPZD, a preprocessor, PRESECOTP2D must first be run to setup all of the input files that are needed by SECOTP2D. This section contains the specific information required to run PRESECOTP2D including the input commands. It also contains speci6c information required to run SECOTP2D. .

7.1 User interactions with PRESECOTP2D

PRESECOTP2D can be exercised interactively (Section 7.1.1) or from a command line (Section 7.1.2). Regardless of the approach, the user must specify up to eight files. A description of the eight files follows:

Files 1-3 and 6 are files that already exist. Files 4, 5, 7 and 8 are created by PRESECOTP2D. Files 4, 5 and 7 are used to run SECOTP2D. As files 1-3 and 6 already exist, they already have assigned names. Files 4-7 and 8 have names assigned by the user. A description of these files follows:

0 File 1 is the input CAMDAT database that contains the grid and properties corresponding to the sampled vector needed for the transport simulation.

0 File 2 is the input CAMDAT database that contains for the sampled vector the source -. information for each species being transported. This file is optional. This information may

also be entered by hand in the PRESECOTPZD input file for simple functions.

File 3 is the ASCII input file that controls PRESECOTP2D. It contains the commands,that establish how SECOTP2D will be run and the information that will go into the input files that are created by PRESECOTP2D. Section 7.2.4 provides a detailed description of all the commands that make up this input file.

Fie 4 is the ASCII input file created by PRESECOTP2D to run SECOTP2D for the sampled vector.

Fie 5 is the b i fie containing the property information used to run SECOTP2D for the sampled vector.

0 File 6 is the optional input velocity file created by POSTSECOFL2D for the sampled vector to transfer double precision velocities fiom SECOFL.2D to SECOTP2D without going through the CAMDAT database which is single precision. This is the preferred way to run SECOTP2D ifthe file is available.

0 File 7 is the binary file containing velocities and source infonnation used to run SECOTP2D for the sampled vector.

File 8 is a diagnostiddebug file containing information about the PRESECOTP2D run. It is - an optional file, but it is necessary that it be speciiied if the user wants to take advantage of the error reporting.

7.1.1 Exercising PRESECOTP2D Interactively

To execute PRESECOTP2D interactively, type PRESECOTP2D followed by a carriage return at the main menu of CAMCON or at the OpenVMS "%" prompt.

A banner scrolls down the screen and then the following information describing the file definitions is printed on the screen.

PRESECOTP2D expects the following files: 1) Input CAMDAT grid and material database 2) Input CAMDAT source database (optional)' 3) PRESECOTP2D mput file 4) SECOTP2D control input file 5) SECOTP2D property data input fde 6) SECOFLZD transfer velocity data file (optional) 7) SECOTP2D velocity data input file 8) PRESECOTP2D diagnostiddebug file (optional)

- The following prompts for filenames appear after the above list of files is p ~ t e d on the

screen The file names in the angle brackets are the defaut names that will be chosen if only a carriage return is entered. Type "cancel" for no file to be chosen when a file name is optional

The Spb0lS PRESECOTPZD$INPUT DIRECTORY and PRESECOTPZDSOUTPUT DIRECTORY

shown below may be assigned with the OpenVMS DEFINE cormnand to be a spec& directory. If they are not dejined, the user must type 1 to specify files in the current directory or [dir-spec], where [dir - spec] is the s p d c a t i o n of a particular directory.

Enter the name of t he Input CAMDAT database <PRESECOTPZD$INPUT-DIRECTORY:CAMDAT.CDB> [Iflow-test .cdb

Enter the name of t h e Input CAMDAT source database (CANCEL f o r no f i l e ) <PRESECOTP2D$INPUTUTDIRECTORY:CAMDATTS.CDB> []source-test.cdb

Enter the name of t h e PRESECOTPZD input f i l e <PRESECOTPZD$INPUT - DIRECTORY:PRESECOTPZD.INP> [IpresecotpZd-test.inp

Enter the name of the SECOTPZD input f i l e <PRESECOTPZD$OUTPUT - DIRECTORY:SECOTPZD.INP> [IsecotpZd-test.inp

E n t e r the name of t he SECOTPZD p rope r ty data inpu t f i l e <PRESECOTPZD$OUTPUT - DIRECTORY:PROPDAT.INP> [lpropdat-test.inp

Enter the name of t he SECOFLZD t ransfer velocity data f i l e (CANCEL f o r no f i l e ) <PRESECOTP2D$INPUT - DIRECTORY:VELOC.TRN> [Iveloc-test . tm

Enter the name of the S E C O T P ~ D veloci ty data inpu t f i l e

30

Enter the name of the PRESECOTPZD diagnostics/debug file (CANCEL for no file) <PRESECOTpZD$OUTPUT-DIRECTORY:PRESECOTPZD.DBG> [)presecotpZd - test.dbg

Ifthe program completes without errors, the message

PRESECOTPZD Normal Completion

appears on the screen. If FORTRAN STOP appears on the screen, an error has occurred. The PRESECOTP2D diagnosticddebug file should be consulted to find a description of the error condition. This can be done using the editor.

7.1.2 Exercising PRESECOTPZD from a Command Line

To exercise PRESECOTP2D %om a command line, type PRESECOTP2D at the OpenVMS prompt, but do not strike the carriage return key. Instead, follow the PRESECOTP2D command with the necessary filenames in the sequence indicated at the beginning of Section 6.0. (Up to 8 filenames are required [see above]; filenames not being used require the qualiiier "cancel") Use the hyphen ("-") at the end of lines on the computer screen as a continuation symbol. The operating system will read it to mean "continued without break on the next line." Thus, although the command procedure is typed on several lines, because of the hyphens, the computer reads it as being typed entirely on one line. - A sample command line procedure that executes PRESECOTP2D could be:

$ PRESECOTPZD flow-test.cdb source-test.cdb presecotpzd-test.inp

- SsecotpZd-test.inp propdat-test.inp veloc-test.trn veldat-test.inp -SpresecotpZd-test.dbg

The command line may also be placed into a command tile and the command file may be executed at the command line or in batch mode. In the command file, the continuation symbol may be used, but the "-%" would not appear at the beginning of the line.

7.2 Description Of Input Files

7.2.1 Input CAMDAT Database F i e

The data required to exist on the input CAMDAT database file is listed below.

Data Name CAMDAT Default symbol Data type

Aquifer properties Aquifer thickness

I- Matrix solid properties (dual porosity)

Porosity

Thick Attniute

Porosity Atkiiute

Tortuosity Tortusty Attribute Retardation factors Rtardd"symbolW property Characteristic length of poros matrix Frctr-Sp property blocks Skin Resistance SkinRest Property

Fracture solid properties Porosity Tortuosity - Dispersivity, longitudinal Dispersivity, transverse Retardation factors

Fporos Attribute Ftortsty Attribute Fdisplng Property Fdisptrn Property

Frtrd-"symbol" Property

Clay solid properties (dual porosity with optional day lining) Porosity Cporos Attribute Tortuosity Ctortsty Attribute Retardation factors Crtrdd"symbol" Property Clay thickness FFilRato . property -

Initial conditions Concenpation Conc-"symbol" Element

Velocity data Cell face centered Darcy x velocity VelFacDX Element Cell face centered Darcy y velocity VelFacDY Element

7.2.2 SECOl?L2D Transfer Velocity Data F ie

This is the input velocity file created by POSTSECOFL2D for the sampled vector. It is used : to transfer double precision velocities fiom SECOFL2D to SECOTP2D without going through

the CAMDAT database, which is single precision.

7.23 Input Source CAMDAT Database

This is an input database that contains the source function. The source function may also entered manually in the input control file (Section 7.2.4).

7.2.4 Input Control F i e

Of the four input files to PRESECOTP2D described in Section 6.0, only one of them is created directly by the user. This file is the PRESECOTP2D ASCII input control file. This file provides input not only to PRESECOTP2D, it is the primary user interface to SECOTPZD also.

This section contains a description of the valid input that can be entered in the PRESECOTPZD ASCII input file. The format of the input file follows the standard format

A

established for CAMCON input files as stated in the User's Reference Manual for CAMCON:

Compliance Assessment Methodology Controller, Version 3.0, section 2.9.4 [23]. The input iile contains keywords, denoted by a leading asterisk (*), and parameters associated with the keywords that can function as secondary keywords or have input values associated with them. Most of the parameters are optional. If a parameter is not entered the default ( in < >) will be taken. If "no defaultn is specified for a parameter, then a value must be entered if that parameter appears in the input iile. The major keywords (denoted with a leading *) must appear on a line by itself The second-level keywords and all of the associated parameters can appear on a single line. The upper case portion of the keywords and parameters shows the minimum character string required to be entered in the input iile.

Twelve keywords are used in the PE'SECOTP~D input Be: (1) *CONTROLgarameters, (2) *VELOCITY-input, (3) * o m m , (4) *-, (5) *SPECIES-data, (6) *PROPERTY-names, (7) %0UNDaryaryconditions, (8) *SOURCE-term, . (9) * IN lT l conditions, (10) *DLSCH~GE-BOUND~, (11) *DP -MESH - (12) *END

The *CONTROL keyword sets up some run control parameters. This keyword is optional. Ifthe keyword is present, any parameter not specified in the input iile will be assigned the d-t value.

type - indicates the type of medium; choices are: SlNmgorosi ty - for single-porosity medium DUALqorosity - for dual-porosity medium

TIME-SCBEME, scheme -GULER>

scheme- indicates the type of time scheme to use; choices are: EULER - Grst order truncation error in time TRAPezodial - second order truncation error in time BACKward - (3-point) second order truncation error in time

SOURCE_COElW, variablenme~=valueI, variable - nameZ=value~, ...

variable m e . is either - AX <0.5> X-coacient of the source terms in the implicit part of the

algorithm; the sum of AX and AY must be 1.

AY <0.5> Y-coefficient of the source terms in the implicit part of the algorithm; the sum of AX and AY must be 1.

function - indicates the type of limiter function to use; choices are: UPWIND - for an upwind scheme CENTERED - for a centered scheme MIxED.WO - for a mixed upwind and centered scheme;

WO sets the portion between the two; W H (upwind); WO=1 (centered)

MINMOD-1 - TVD limiter function; minrnod(1,r); very dissipative MINMOD-2 - TVD limiter function; minmod(l,2r) MINMOD 3 - TVD limiter function; minmod(2,r) MINMoDI~ - TVD limiter function; minmod(2,2r) MVSCL - TVD limiter function; van Leer's MLTSCL SOBERBEE - TVD limiter function; Roe's superbee; highly compressive

name - name of the CAMDAT variable containing the climate multiplier, must be a valid CAMDAT element variable name

The *VELOCITY keyword controls how the velocities are read fiom the CAMDAT database. If the velocities are being read &om the input velocity file generated by POSTSECOFL2D, this keyword is omitted. If the keyword is present, any parameter not specified in the input file will be assigned the default value.

X-DARCY, name <VELFACDD

name - narne of the CAMDAT variable containing the x-component of the face centered Darcy velocity; must be a valid CAMDAT element variable narne

Y-DARCY, name CVELFACDP

name - name of the CAMDAT variable containing the y-component of the face centered Darcy velocity; must be a valid CAMDAT element variable name

FLOW-CODE, source <SECOV

source - identifies if the flow calculation was done the SECOFL2D code or another flow code to determine how the velocities for the ghost cells should be set up. Ifthe flow calculation was done by the SECOFL2D code, these velocities get recorded on the database; otherwise, they are simply obtained from the nearest neighbor. Choices are: SECO2 - for the SECOFL2D flow code OTHER - for any other flow code calculation

- STEP, step <I>

step - the step number to start reading the velocities from the CAMDAT database or the velocity transfer fle. This number must be greater than 0 and must not exceed the number of steps on the database or the velocity transfer file.

STEADY, s t e 4 <NO>

stendy - indicates ifthe flow field is steady state; choices are YES or NO; the STEP keyword can be used to select the desired time step to use the steady state . --.. . *

velocity. i

"OUTPUT

The *OUTPUT keyword controls how the output is written to the binary output fie and to the terminal. This keyword is optional. If the keyword is present, any parameter not spedied in the input fle will be assigned the default value.

STEP, step <I>

step - output printout control parameter for concentration; entering a value of n means that computed values of concentration are to be printed at every nth time step; must be great& than 0 and less than or equal to the number of time steps selected (see *TIME, NUMstep)

S(SREEN-10, mode <OFF>

mode - enables or disables iteratioh information to be written to the terminal during the SECOTP2D run; choices are: ON or OFF

The *TIME keyword sets up the time interval and the time steps for the SECOTPZD run. All of these parameters are considered by the code as optional, but in reality some of them are not. The user needs to choose ippropde values for setting up the time steps, start and stop h e , etc. as desired.

NUM-step, steps <O>

steps - number of time steps for the transport calculation (does not have to match the number of steps on the velocity database as interpolation is done automatically); must be greater than 0.

TIME-GENeration, mode <AUTO>

mode - Specifies the method of time step generation; choices are: AUTO - generates the time step using the equation

(STOP-TIME - START-TIME)/(NUM-STEP - 1) MANual - uses the specified DELTA-T for the time step; if

DELTA-T"NUM-STEP is greater that STOP-TIME, the simulation ends at STOP-TIME; $DELTA T*NUM-STEP is less than STOP-TIME, the simulation ends at DELTA-T*NUM-STEP

VAR - indicates that the time step is variable and will be calculated using the parameters entered with the VARDT keyword; delta - t(l)=INlT.-DT, delta-t(n) = (delta-t((n-l)*GROW-FAC,MAX-DT)

START - TIME, time <O.O>

time - initial time; must be equal to or greater than 0,

STOP - TIME, time <O.O>

time - end time; must be greater than 0.

varidle name, is either -

I?SI-DT <O.O> - initial delta t; must be greater than 0.

GROW-FACtor <0.0> - delta t multiplier (growth factor); must be greater than 0.

MAX-DT <O.O> - maximum value for delta t; must be greater than 0.

The *SPECIES keyword sets up the radionuclides that will be transported. At least one radionuclide and at least one chain must be set up using these parameters. The default values are set to 0, but this value is never valid for any of the parameters. All of the parameters should appear in the input iile for each radionuclide being set up.

variable - name, is either

SYMBOL <no defaulv - name of the nuclide; usually the fist one or two characters are the chemical symbol and the last three characters are the atomic mass unit of the isotope; only the leading alpha characters are used to match "symbol" for the retardation property names on the CAMDAT

INDEX

LAMBDA

database; any alpha numeric characters are valid; the - length must not exceed 5.

<0> - component number of the nuclide; must be greater than 0.

<0> . - decay constant of the nuclide ( W t m ) ; can be either a value or a CAMDAT database variable name containing the decay constant; i fa value is entered it must match the number on the Secondary Database

FREE-H20-DIFF <O> - h e water molecular dii%sion; used with the tortuosity to calculate the molecular diision; can be either a value or a CAMDAT database variable name containing the decay constant; i fa value is entered it must match the number on the Secondary Database

CURIE <0> - conversion &tor for converting mass flux to activity (Cukg); can be either a value or a CAMDAT database variable name containing the decay constant; if a value is entered it must match the number on the Secondary Database

CHAWCHAWdefiition, variable - namel =valuel, variable - name2=value2, ...

variable name,, is either -

CHALN-NUMber <0> - chain number, must be greater than 0; chain number must be unique.

NUM-SPECIES <0> - number of species in this chain; must be greater than 0.

NUC-INDICES <0> - list of indices (from INDM above) of the species in this chain in proper ordw, must all be greater than 0.

The *PROP keyword establishes what CAMDAT variable names contains the material properties of the transport medium. All of these parameters are optional. If any of them are not specified in the input ae, the default value will be taken. If the default value is used, that name must be a valid name on the CAMDAT database.

- AQUIFER, variable - mel=value, , variable - name~=valuez, ...

variable-name" is either

THICKness m C K > - CAMDAT attribute name for aquifer thickness; must be a valid CAMDAT variable name

DIFFusive, variable - name~=valuel, variable - name2=valuea ...

0 variable name,, is either -

TORT (TORT> - CAMDAT attribute name for the diffusive tortuosity of the matrix material; must be a valid CAMDAT variable name

POROSITY eOROSITY> - CAMDAT attribute name for the diffusive porosity of the matrix material, must be a valid CAMDAT variable name

RETARD-factor ~TARDARD"symbol"> - CAMDAT property name for the diffusive retardation fkctor of the nuclide with the chemical symbol "symbol" in the porous matrix material; the default name is automatically generated using the nuclides listed on the NUClide cards under the *SPECIES keyword; must be a valid CAMDAT variable name

DUALqorosity, variable-name~=value,, variable - name~=value~, ...

variable name, is either -

BLOCK-LENgth IFRCTR-SP> - CAMDAT property name for the half length of the porous matrix bloch for a dud porosity medium; must be a valid CAMDAT variable name

- CAMDAT property name for the skin resistance; must be a valid CAMDAT variable name

CLAY-THICKness CFFILRATO> - CAMDAT attribute name for the ratio of - the clay thickness to the firacture aperture; must be a valid CAMDAT variable name

variable - name, is either

DISP - LNG

DISP - TRN

TORT

POROSITY

RETARD-factor

<FDISPLNG> - CAMDAT attribute name for longitudinal dispersivity, must be a valid CAMDAT variable name

eDISPTRN> - CAMDAT attribute name transverse dispersivity; must be a valid CAMDAT variable name

cFTORT> - CAMDAT attribute name for the diffusive tortuosity of the eactured material; must be a valid CAMDAT variable name

<FPOROS> - CAMDAT attribute name for the a i v e porosity of the fractured material; must be a valid CAMDAT variable narne

4XIX.D-"symbol"> - CAMDAT property name for the diffusive retardation factor of the nuclide with the chemical symbol "symbol" in the fractured material, the default name is automatically generated using the nuclides listed on the NUClide cards under the *SPECIES keyword; must be a valid CAMDAT variable name

CLAY-LINING, ~ar iab le~rne~-Yalue~ , variable - name2=valuea ...

variable name, is either -

TORT -=TORTCLAY, - CAMDAT aasibute name for tortuosity of the clay lining; used only for dual porosity models; must be a valid CAMDAT variable name

POROSITY *ORCLAP - CAMDAT attribute name for the porosity of the clay lining; must be a valid CAMDAT variable name

RETARD-factor 4XTDC-"symboln> - CAMDAT property name for the retardation factor of the nuclide with the chemical symbol "symbol" in the clay lining; the default name is automatically generated using the nuclides listed on the WClide cards under the *SPECIES keyword; must be a valid CAMDAT variable name

The *BOUND keyword sets up the boundary conditions for the SECOTP2D run. This keyword is optional. Ifit is not included, the default of "AUTO" is set for all species and all boundaries. If the keyword is present, any parameter not specified in the input file wiU be assigned the defaut value. Any parameier that lias no default designated is not optional and must be included.

TOP, variable - namel=vaIuel, variable-name2=value~ ... - vmiabk - name, is either

TYPE <AUTO> - type of boundary condition; choices are: AUTO - SECOTP2I) automatidy selects the

type based on the velocity field N E W - for a Neumann boundary condition DIRichlet - for a Dichlet boundary condition

WCR

VALUE

SYMBOL

NRANGE

<O.O> - value of concentration or gradient; must be equal or greater than 0

<no defaut> - name of the nuclide associated with the specified concentration; must be a valid symbol name

'<O,O> - range of elements for which this boundary condition applies; must be greater than 0 and less than or equal to number of elements in the x-direction

- increment for element range in NIMNGE; must be greater than 0 and less than or equal to the number of elements in the x-direction

BOTtom, variable-name~=valuel, variable - name2=value2, ...

0 variable name, is either -

TYPE <AUTO> - type of boundary condition; choices are: AUTO - SECOTP2D automatically selects the type

based on the velocity field NEUMAN - for a Neumann boundary condition DIRichlet - for a Dichlet boundary condition

VALUE <O.O> - value of concentration or gradient; must be equal or greater than 0

SYMBOL <no default> - name of the nuclide associated with the speciiied concentration; must be a valid symbol name

NRANGE <O,O> - range of elements for which this boundary condition applies; must be greater than 0 and less than or equal to number of elements in the x-direction

INCR <I> - increment for element range in NR4NGE; must be greater than 0 and less than or equal to the number of

-. elements in the x-direction

0 variable me. is either -

TYPE <AUTO> - type of boundary condition; choices are: AUTO - SECOTPZD automatically selects the type

based on the velocity field NEUMAN - for a Neumann boundary condition DlRichlet - for a Dirichlet boundary condition

VATSJE

SYMBOL

NRANGE

<O.O> -value of concentration or gradient; must be equal or greater than 0

<no default> - name of the nuclide associated with the specified concentration; must be a valid symbol name

<O,O> - range of elements for which this boundary condition applies; must be greater than 0 and less than or equal to number of elements in the x-diuection -.

42

INCR - increment for element range inNR4NGE; must be greater than 0 and less than or equal to the number of elements in the x-direction

RIGHT, vm'able6namel=valuel, vmia6le-name~=value~, ...

vari'able name, is either -

TYPE <AUTO> - trpe of boundary condition; choices are: AUTO - SECOTP2D automatically selects the type

based on the velocity field NEUMAN - for a Neumann boundary condition DIRichlet - for a Dirichiet boundary condition

VALUE <O.O> - value of concentration or gradient; must be equal or greater than 0

SYMBOL - <no default> - name of the nuclide associated with the specified concentration; must be a valid symbol name

NRANGE <O,O> - range of elements for which this boundary condition applies; must be greater than 0 and less than or equal to number of elements in the x-direction

INCR <I> - increment for element range in NRANciE., must be greater than 0 and less than or equal to the number of ...-.,

,,"l

elements in the x-direction ?.,!:, '\ i . .... ,., , , . .. i I r, G-"$ i : . I. (:\, 1 , ; .' . ,' ,, '-; < ~ , , ,.,:

*SOURCE-term ? . :~ . '--...~~ ~.... ,

The *SOURCE keyword sets up the source definition for each radionuclide. A source function must be set up for each radionuclide either by obtaining it &om the source database or by manually setting up the times and function values. The function may be identically zero.

SOURCE, source

source - set to CAMDAT ifthe source term times and function values are to be read fiom the history variables of the source CAMDAT database; there are no other choices.

TERM - DEFiition, variable - namel=valuel, variable - name2=value2, ...

.-, variable name. is either -

A,.

SYMBOL <no defaulv - name of the nuclide for which the source term is specified; must be a valid symbol name

NAME-SOL <no defaulv - name of CAMDAT history variable that contains the integrated source term information; must be a valid CAMDAT variable name

NAME-FLUX <no default> - name of CAMDAT history variable that contains the source term flux information; must be a valid CAMDAT variable name

VALUES <O.O>

]RANGE <O,O>

JRANGE <O,O>

- number of control points on the graph of the function versus time; or the number of time steps in the source CAMDAT database; must be greater than 0

- array of time values corresponding to control points on the graph of function versus time; first time must be greater than simulation start time; not required ifthe source data is read fiom a CAMDAT database

- array of integrated function values corresponding to the TIMES; a l l values must be greater than or equal to 0; not required ifthe source data is read fiom a CAMDAT database

- range of elements in the i-direction to apply the source tern described in this definition; must be greater than 0 and less than or equal to-the number of elements in the x direction.

- range of elements in the j-direction to apply the source term described in this definition; must be greater than 0 and less than or equal to the number of elements in the y direction.

The *INIT keyword sets the initial conditions for the radionuclides. This keyword is optional. I€ it is not presenf the initial concentration of each radionuclide is set to 0. If the keyword is present, any parameter not speciiied will in the input me be assigned the default value. Any parameter that has no default designated is not optional and must be included.

/-->;-\ / r *

i ' r '

TIME-STEP, time <0.0>

time - time step f?om CAMDAT to read the initial conditions; must be greater than 0 and less than or equal to the number of steps on the input CAMDAT database

DEFition, variable-mel=valuel. variable - me2=value2, ...

variable - name, is either

NAME

INCrement

W G E

SYMBOL

VALUE

<CONC-"symbol"> - CAMDAT element variable name storing initial value of concentration for the nuclide s p d e d by "symbol"; the default name will be automatically generated using the nuclides listed on the NLTClide cards under the *SPECIES keyword; must be a valid CAMDAT variable name.

- element increment for elements specified by NRANGE; must be greater than 0 and less than or equal to the number of elements

- range of elements where the value CONC applies; must be greater than 0 and less than the number of elements

<no default=- - name of the nuclide for which the initial value of CONC or DEF is s p d e d , must be a valid symbol name

- concentration value used to overwrite the CAMDAT value or used for nodes d&ed by NRANGE; must be greater than or equal to 0

The *DISCHARGE-BOUND keyword sets up rectangular discharge boundaries around which the source is integrated. This keyword is optional. If it is specified in the input file, appropriate values must be entered for each of the parameters. " - ./ ---

- N[TM-BNDS, m b e r <O>

mmrber - number of discharge boundaries to set up; must be greater than or equal to 0 -

0 variable name, is either -

TOPLEFT <0,0> - i, j coordinate of top left comer of the rectangular discharge boundary; must be contained in the grid

BOTTOM-RIGHT <0,0> - i, j coordinate of bottom right comer of the rectangular discharge boundary; must be contained in the grid

The *DP MESH keyword is used to enter data describing the discretization of the porous matrix blocks used for dual porosity. This keyword must be present for dual porosity runs. It will not be present for single porosity d.

AUTO, variable - namel=vaIuel, variable-name~=vaIue~, ...

0 variable - name, is either

INlT-DIST <O.O> - initial non-dimensional cell size at the matrix hcture interface; must be greater than 0 and less than 1

NUM-NODES <O> - number of one nodal points in each grid line representing the porous matrix domain; must be greater than 0

CLAY-FRAC <0.0> - initial cell size for fractures with a clay lining; units are a fraction of the clay lining (i.e., .3, .IS, etc.); must be greater than 0 and less than 1

MANu-4 variable - namel=valuel, variable-name2=vaIue2, ...

0 vmiable name, is either -

NUM-NODES <O> - number of one dimensional nodal points in each grid line representing the porous matrix domain; must be greater than 0 f"" -

b-

DISTances <O> - list of nodal values of dimensionless distance from the center of the matrix block this dimensionless distance is defined as the ratio of the actual distance coordinate to the half-thickness of the block all must be less than 1 and monotonically increasing.

The *END keyword signals the end of the input Be. There are no other keywords or parameters associated with it.

SUMMARY

The following summarizes all PRESECOTP2D input keyword and parameters:

3. SOURCE-COEFF, AX=, AY=

1. STEP-

*TIME

1. NUM_step=

2. TIME - GENeration=

3. DELTA-T=

4. START-TIME=

5. STOP - TIME=

6 . VAR - DT, INIT - DT=, GROW - FACtor=, MAX - DT=

*SPECIES - data

1. NLTClide, SYMBOL=, INDEX=, LAMBDA=, FREE - H20_DIFF=, CUKE=

2. CHAIN-definition, CHAIN-NUMber=, NOM-SPECIES=, NUC-INDICES=

*PROPerty-names

1. AQUIFER, THtCKness=

2. DIFFusive, TORT=, POROSITY=, RETARD-factor=

2. DUALqorosity, BLOCK-LENgth=, SKIN-RESIStance=, CLAY_THICKness=

3. ADVECtive, DISP - LNG=, DISP-TRN=, TORT=, POROSITY=, RETARD-factor=

4. CLAY - LINING, TORT=, POROSITY=, RETARD-factor=

*BOU'N.Daty-mnditions

1. TOP, TYPE=, VALUE=, SYMBOL=, NRANGE=, WCR=

2. BOTtom, TYPE=, VALUE=, SYMBOL=, NRANGE=, INCR= -,

48

3. RIGHT, TYPE=, VALUE=, SYMBOL=, NRANGE=, INCR=

4. LEFT, TYPE=, VALUE=, SYMBOL=, NRANGE=, WCR=

...------ ~

*EM> J ,- ," .. , . ,,, . . .... * * *

7.3 Input Fie for PRESECOTPZD

The following is an example input file for PRESECOTP2D.

! ! CHAIN 1 PU240 - ! CHAIN 2 PU239 ! CHAIN 3 AM241 -> NP237 -> U233

! CHAIN 4 U234 -> TJ3230 ! ! *CONTROL

MEDIUM=SINGLE Tlh4E-SCHEMEMETRAP

! *OUTPUT

STEP=lOO ! *TIME

NUM_STEPS=1000 START TIME=3.155 8E+ 10 STOP-?~ME=~. 1558~+l l

! *SPECIES

NUCLIDE, SYMBOL=PU240, INDEX=l, LAMBDA=3 36E-12, FREE-H20 DIFFq.48E-10

NUCLIDE, sYMBoL=%JZ~~, INDEX=2, LAMBDA=9.13E- 13, FREEFREEH20-DIFFq.48E-10

NUCLIDE, SYMBOL=AM241, INDEXE3, LAMBDA4.08E-11, FREE-H20-DIFF=l. 76E- 10

NUCLIDE, SYMBOL=NP237, INDEX4, LAMBDA=l.O3E-14, FREE-H20 DIFFz1.76E-10

NUCLIDE, sYMB0~=6233, INDEX=5, LAMBDA=1.39E-13, FREE-320 DIFF=1.7OE-10

NUCLIDE, SYMBOL=^^^^, INDM=6, LAMBDA=8.98E-14, FREE H20 DIFF=2.7OE-10

NUCLIDE, SYGOL=-=O, INDEX=7, LAMBDA=2.85E-13, FREE-H20-DIFF=l .OOE-10

I CHAIN 1 PU240 I CHAIN 2 PU239 ! CHAIN 3 AM241 ->NP237 -> U233 ! CRAW 4 U234 ->TH230

CHAIN CHAIN-NUM=l, NUM SPECIES=l, NUC-INDICES=l CHAIN CHAIN-NUM=2, NUM~SPECIES=I, NUC-INDICES=2 CHAIN CHAIN-NUM=3, NUMNUMSPECIES=3, NUCCINDICES=3,4,5 GRAIN CRAW - NUM=4, NUM - SPECIES=2, NUC-INDICES=6,7

I

*PROPTERTY-NAMES ADVECTIVE TORT=FTORT, POROSITY=FPOROS, RETARD-FRTRD

I

*SOURCE-TERM SOURCE = CAMDAT TERM DEF NUM_POINTS=192, NAMJ-SOL= MOOPU240, SYMBOL=PU240, &

~ ~ ~ = 1 3 , 1 3 TRANGE=43,43 -

- TERM DEF NUM_POINTS=192, NAME-SOL= MOOPU239, SYMBOL=PU239, & @GE=13,13 JRANGE=43,43

TERM-DEF NUM-POlElTS=192, NAME-SOL= MOOM41, SYMBOL-M41, & IRANGE=13,13 JRANGE=43,43

TERM DEF NUM_POINTS=192, NAME-SOL= MOONP237, SYMBOL=NP237, & @GE=13,13 IRANGE43,43

TERM DEF NUM POINTS=192, NAME-SOL= MOOU233, SYMBOL=U233, & @GE=l3, 13 -%NG~=43,43

TERM-DEF NUM-POlElTS=I92, NAME-SOL= MOOU234, SYMBOL=U234, & IRANGE=13,13 IRANGE=43,43

TERM DEF NUM POINTS=192, NAME - SOL= MOOTH230, SYMBOL=TH230, & @~~=13,13-IRANGE=43,43

*END

7.4 INPUT PARAMETER CHECKING

All of the keywords and parameters entered in the PRESECOTP2D ASCII input file are checked by PRESECOTP2D-for validity. The input values associated with the parameters are not all checked for validity. The values that are checked by the code will cause the execution to abort if an invalid value is entered. An error message is written to the diagnostiddebug output iile

A

describing the parameter and the value that generated the error.

The following is a list of the input values that are not checked by the code for a valid range.

*CONTROL-PARAMETERS SOURCECEcOEFF, AX =

AY = *OUTPUT

STEP =

*TIME NUM-STEP =

DELTA-T = START-TIME = STOP-TIME = VAR-DT, INIT-DT =

GROW-FACTOR = MAXDT =

* SPECIES-DATA CHAIN - DEFINITION, CHAIN-NUMBER =

NUM-SPECIES = NUC-NDICES =

*BOUNDARY~CONDmONS TOP, VALUE =

NRANGE =

INCR = BOT, VALTJE =

NRANGE = INCR =

RIGHT, VALUE NRANGE =

INCR =

LEFT, VALUE = NRANGE = INCR =

* SOURCE-TERM TERM - DEFINITION, NUM_POINTS =

TIMES = VALUES = IRANGE = JRANGE =

CONDITIONS TIME:sTEP =

D-ON, IM= = .~ NRANGE =

VALUE =

'DISCHARGE-BOUNDARIES NUM-BNDS = BOUNDUM>DERNITION, TOP-LEFT =

BOTTOM-RIGHT = *DP-MESH

AUTO, m - D I S T =

NUM-NODES = CLAY-FRAC = ,/ .:-.,.,,

-,~, MAN, NUM-NODES = "9 '*.,,% ',

DISTANCES = I ,+.. > , . + a * . I ? >

*t .' 'a- ?.\. , . , y ,

7.5 User Interactions With SECOTP2D 4:;: ",,.,, :' i: \LJ" . , . ' . ' , : .

To execute SECOTP2D for a sampled vector, type SECOTP2D at the Alpha system "%" prompt. SECOTP2D will request the names of three files. Alternatively, the user may append the names of the three files (in the order listed below) to the SECOTP2D command line. The three files that the user must speafy are listed below:

1. The input control file. This file specifies the processing options for SECOTP2D. Unlike the input control files for most of the WIPP PA codes, the user has no interactive control over the contents of this file. This iile is generated by the upstream code PRESECOTP2D. AU of the interaction with this file is done through choices made in the input file to the preprocessor. This file in turn contains the names of the binary property and velocity files needed to m this code (see below).

2. The binary output data file. This file contains the output of SECOTP2D in binary format. This file is processed by POSTSECOTF'2D.

3. The diagnosticddebug file. This file contains run-time information on the execution of SECOTPZD.

In addition, two other input files are required for SECOTP2D to run. These files, listed below, are specified by the user when the preprocessor is run.

0 Property data input file. This file contains the property information output fiom PRESECOTP2D in binary fofmat, which is used to run SECOTP2D.

Velocity data input file. This file contains the velocities and source information output fiom PRESECOTP2D in binary fon-nat used to run SECOTP2D.

7.6 Description Of Input F i

7.6.1 Input control F i e

The input control file to SECOTF'2D is generated by PRESECOTP2D based in part on its own input control file. For all practical purposes, the code-generated input control file to SECOTPZD

I- is transparent to the user.

7-63 Prop- Data Input and Velocity Data Input Fies

The property data input file, which contains CAMDAT property information, is binary and is not user-spded. It is therefore transparent to the user. The velocity data input file, which contains CAMDAT velocities and source information, is b i and is not user-specified. It is therefore transparent to the user.

7.7 Description Of Output Files

7.7.1 Binary Output Fie

The binary output file contains the output of SECOTP2D in binary format. Because it is binary, it cannot be read by the user.

7.73 DiagnostidDebug Output F l e

A sample diagnosticddebug file, the only output file fiom SECOTP2D that can be read by the user, is provided in Appendix IL

-. 7.8 Postprocessor, POSTSECOTPZD ', ' ,

. . > : ,

POSTSECOTP2D is a postprocessor that writes SECOTPZD output to a CAMDAT database. It - writes up to three types of data to the database. These are 1) face centered Darcy velocities (specific discharges) that were used by the simulation, 2) concentration at each element in the grid for each radionuclide transported, and 3) discharge information for each radionuclide transported and each discharge boundary set up. Discharge boundaries are optional and this information is written if one or more discharge boundaries are set up. The following is a list of the input and output files that are needed to run POSTSECOTP2D.

1. Input CAMDAT database 2. SECOTP2D binary output file 3. Output CAMDAT database 4. POSTSECOTP2D diagnostiddebug file (optional)

The first file is the input CAMDAT database that was used by PRESECOTP2D. It contains property and grid information. The second fie is the binary output file fiom SECOTPZD. The third file is the output database created by POSTSECOTP2D. It wiU contain the information from the input database and also the results %om SECOTP2D. The fourth file is an optional diagnostiddebug file created by POSTSECOTP2D that contains information and error messages generated by running POSTSECOTPZD. POSTSECOTPZD does not require an ASCII input command file as did PRESECOTP2D.

8 TEST PROBLEM

To run SECOTP2D in the CAMCON environment, first a preprocessor, PRESECOTP2D, must be run Then at the end of the calculation a postprocessor, POSTSECOTP2D, must be run. The preprocessor creates the necessary input files for SECOTP2D by translating data fiom an input CAMDAT database to b i property and velocity files. The postprocessor creates an output CAMDAT database by addiig the results of the run to the input CAMDAT database. The following test problem gives an example of how to run each of the separate three codes.

This test problem is one of the 1992 WJPP PA vectors [17]. It is discretized by a 46x53 grid. The input velocity field was generated by SECOFL2D.

The purpose of running PRESECOTP2D is to produce the necessary input files so that SECOTP2D may be run. To run PRESECOTP2D, CAMDAT databases are needed as well as an ASCII input file.

A copy of the PRESECOTP2D ASCII input file, PRESECOTP2D TEST.INT follows. VELOC TEST.TRN is an input velocity file that was created by POSTSECOK2D. The binary files created by PRESECOTP2D, PROPDAT-TEST.INP and VELDAT-TEST.INP cannot be printed, but can be found in the directory CAMCONSROOT: pRESECOTF'2D .TESTI.

The test case for PRESECOTP2D can be run on the Alpha (Beatle) in the directory user's the following file assignments:

Input CAMDAT database: CAMCON%ROOT:~RESECOTP2D.TESTlFLOW_TEST.CDB

Input CAMDAT source database: CAMCON%ROOT:~RESECOTP2D.TESTJSOURCEETEST.CDB

PRESECOTP2D input filename: CAMCON$ROOT:pRESECOTP2D.TES~PRESECOTP2D-TEsT.m

SECOTP2D control input filename: SECOTP2D-TEST.INP

SECOTP2D property data input filename: PROPDAT-TEST.INP

', SECOFL2D transfer velocity data filename:

CAMCON$ROOT:~RESECOTP2D.TEST]VELOCCTEST.TRN

SECOTPZD velocity data input filename: VELD AT-TEST.INP

PRESECOTP2D diagnosticddebug filename: PRESECOTP2D-TEST.DBG

! Test case for SECOTP2D

*CONTROL MEDIUM=dual TIME-SCHEME=back LZMITER=muscl

*VELOCITY s m = 1

*OUTPUT STEP=50 SCREEN-IO=ON

*TIME NUM STEP=500 TR&-GEN=AUTO START-TIME=~ .~~~~~EIO STOP TIME =3.15569Ell

*SPECIES NUCLIDE SYMBOL=U233, INDEX=l, LAMBDA=1.39E-13, &

FREE F320 DIFF=1.70E-10, CUR.lE=9.68 CJiIATN C---NUM=l NUM_SPECIES=l NUC-INDICES-1

*PROPERTY DEITJS, TORT= MTORT, POROSITY=MPOROS, RETARD=ZRTRD DUAL BLOCK LEN=BLOCIU.EN SKINKINRESIST= PZERO ADVECTIVE ~ISP-LNG-DIP-LNG, DISP-TRN=DISP-TRN, TORT= FTORT, &

POROSITY=FPOROS, RETARD=ZRTRD *SOURCE

SOURCE = CAMDAT TERM-DEF SYMBOL=U233, NM-SOL=MOOU233, NUM POINTS= 186, &

IRANGE= 21,21, TRANGE=45,45 *DISCHARGE-BOUND

NUM-BNDS=3 BOUND DEF TOP_LEFT=1,53, BOTTOM-RIGHT=46,3 BOUND-DEF TOP LEFT=1,53, BOTTOMEFRIGHT=46,23 BoUNDDEF TOP-LEFT=~,~~, - BOTTOM - RIGHT=46,34

*DP-MESH- AUTO INIT-DIST=.OI, NUM-NODES=lO

*END

This test case for SECOTP2D was chosen to exercise features of the code that are used while running SECOTP2D for the PA calculation. This example exercises the dual porosity capabiity of the code. There is one chain with one species. The binary files, PROPDAT-TEST.INF' and VELDAT-TEST.WP, are input files. The b i i files SECOTP2D-TESTBlN, and the ASCII file, SECOTP2D-TEST.OUT, are output mes for SECOTP2D. The input files can be found in the directory CAMCONSROOT:[SECOTP2D.TEST].

The test case for SECOTP2D can be run on the Alpha (Beatle) in the directory using the following file assignments:

SECOTP2D input filename: CAMCON%ROOT:[SECOTP2D.TEST]SECOTP2D-TEST.INP

SECOTP2D binary output filename: SECOTP2D-TEST.BIN

ENTER SECOTP2D output filename: SECOTP2D-TEST.OUT

The purpose of running POSTSECOTP2D is to add the results fiom a SECOTP2D run to a CAMDAT database. The output results file is b i as are the CAMDAT databases. The data on the CAMDAT database may be examined or plotted using BLOT [22]. The test case for POSTSECOTP2D can be run in the user's directory using the following file assignments:

Input CAMDAT database: CAMCON%ROOT:~OSTSECOTP2D.TESTJFLOW TEST.CDB

SECOTP2D b i i output filename: SECOTP2D-TESTBIN

Output CAMDAT database: SECOTPZD-TEST.CDB

POSTSECOTP2D diagnosticddebug filename: POSTSECOTP2D-TEST.DBG

A plot for this example showing the contours of concentration for COW233 at TIME = 10,000 years, created using BLOT, is shown in Figure 26.

REFERENCES

[I] Roache, P. I., 'Computationd Fluid Dynamics Algorithms Developed for WIPP Site Simulations,' Proceedings IX International Conference on Computational Methods in Water Resources, Denver, Colorado, 9-12 June 1992, T. Russell, et al., eds., pp. 375-382.

[2] Roache, P. J., 'Computational Fluid Dynamics Algorithms and Codes Developed for WIPP Site S i t i o n s , ' Proceedings Asian P a 5 c Conference on Computational Mechanics, 11-13 December 1991, University of Hong Kong, J.H.W. Lee, et al., eds., H. Balkeema, Amsterdam.

[3] Salari, K, Knupp, P., Roache, P., and Steinberg, S., 'TVD Applied to Radionuclide Transport in Fractured Porous Media,' Proceedings M International Conference on Computational Methods in Water Resources, Denver, Colorado, 9-12 June 1992, T. Russell, et al., eds., pp. 141-148.

[4] Roache, P. J., 'The SECO Suite of Codes for Site Performance Assessment,' Proceedings 1993 I d Rtgh-kel Radi.oactive Waste Management Cod, April 26-30, 1993, Las Vegas, NV.

[5] Huyakom, P. S. and Pinder, G. F., Computational Methodr in Subsu@ace Flow, Academic Press, New York, 1983.

-.

[q Huyakom, P. S., Lester, B. H, and Mercer, J. W. 'An Efficient Finite Element Technique for Modeling Transport in Fractured Porous Media: S i e Species Transport,' Wder Res. Res., VoL 19, No. 3, pp. 841-854, 1983.

[q Bear, J. and Bachmaf Y. Introduction to Modeling of Tramport Phenomena in Porous Merfia; Kluwer Academic Publishers, Dordrecht, Netherlands, 1990.

[8] Scheidegger, A E, The Physics of Flow Through Porous Meda, University of Toronto Press. 1974.

[9] Streltsova-Adams, T. D., 'Well Hydraulics in Heterogeneous Aquifer Formations,' Advrmces in Hy&osn'ence, Vol. 11, (Ed., Chow, V.T.), pp. 357423 Academic Press, New Yo* 1978.

[lO]Fletcher, C. A J., Compum~onal Techniques for Fluid Dynmnics, Vol. I and lT, Springer- Verlag, 1988.

[11]Sweby, P K , 'High Resolution Schemes Using Flux L i t e r s for Hyperbolic Conservation Laws', SL4MJ Numer. Anal. 21: 995-101 1.

[12]Yee, H C., 'Construction of Explicit and Implicit Symmetric TVD Schemes and Their Applications,'J Comp. Phys., Vol. 68, pp. 151-179, 1987.

.-

58

h

[13]Tang, D. H., Frind, E. O., and Sudicky, E. A 'Contaminant Transport in Fractured Porous Media: Analytical Solution for a Single Fracture,' Water ResourcesResearch, Vol. 17, No. 3, pp. 555-564, 1981.

[14]Lester, D. H., Jansen G., and Burkholder, H. C., 'Migration of Radionuclide Chains Through an Adsorbing Medium,' AIChE Symposium series, Vol. 71, No. 152.

[15]Roache, P. J., 'A Method for Uniform Reporting of Grid Refinement Studies,' ASME FED- Vol. 158, Quantification of Uncertainty in Computational Fluid Dynamics, ASME Fluids Engineering Division Summer Meeting, Washington, DC, 20-24 June 1993. Celik., I, Chen, C. J. Roache, P. J., and Scheurer, G. Eds., pp. 109-120.

[16]Roache, P. J., 'A Method for Uniform Reporting of Grid Refinement Studies,' Proc. 11th AIAA Computational Fluid Dynamics Conference, July 6-9, 1993, Orlando, Fl., Part 2, pp. 1- 57-1058.

[17]Roache, P. J., 'Perspective: A Method for Uniform Reporting of Grid Refmement Studies,' ASME Journal ofFluids-Engineering, Vol. 116, No. 3, Sept.1994, pp. 405413.

[18]Anon., 'Preliminary Pdonnance Assessment for the Waste Isolation Pilot Plant,' Sandia Report SAND92-070014, December 1992. -

[19]Sudicky, E. A and Frind, E. O., 'Contaminant Transport in Fractured Porous Media: Analytical Solutions for a System of Parallel Fractures,' Wofer Resources Research, Vol. 18, NO. 6, pp. 1634-1642, 1982.

[20]Davis, G. B. and Johnston, C. D., Comment on "Contaminant Transport in Fractured Porous Media: Analytical Solutions for a System of Parallel Fractures" by Sudicky, E. A, and Frind, E. O., WaterResourcesResemch, Vol. 20, No. 9, pp. 1321-1322, Sept. 1984.

[2l]van Gulick, P., 'Evaluation of Analytical Solution for a System of Parallel Fractures for Contaminant Transport in Fractured Porous Media,' Sandia Report SAND 942877, to appear. Submitted for publication to Water Resources Research.

[22]Javandel I., Doughty, C., and Tsang, C. F. Groundwater Trancpon: Handbook of MathematicaIModels, American Geophysical Union, Washingtoq D.C., 1984.

[23]Rechard, R P., ed., 'User's Reference Manual for CAMCON: Compliance Assessment Methodology Controller Version 3.0,' Sandia Report SAND90-1983 UC-721, August 1992.

APPENDIX 2D Transport Benchmark Solutions - I

P. Kdupp Ecodynamics Research Associates, P.O. Box 9229, Albuquerque, NM 87 1 19

The following tests the basic 2D transient algorithm for hc ture transport with advection, diffusion, and decay.

The governing equation for this benchmark is:

where

This equation is obtained fiom the more general equation solved in the transport wde by - assuming

Free-water molecular diffusion is zero, Dual porosity term is off, Single-species decay.

One way to construct a solution which has time-independent boundary conditions is to assume the concentration is of the form

with CM a constant. This is a solution provided

To attain time-independent Dichlet boundary conditions fiom (3), we must have p = 0 on the boundary of the domain. Assume the domain is f2 = [0, L]' and let

xx xy p(x, y ) = sin-sin-. L L

A

To evaluate g explicitly, expand (4):

The partial derivatives ofp are:

x m . I y p, = -cos-sm-, L L L x . m I y

p, = -sm-cos-, L L L

x2 p =-- = ~2 P>

x2 PW = - ~ z p .

xZ I y p, = - a s - - w s -

L2 L L

The derivatives of the Dispersion Tensor are obtained fiom (2):

fork= 1, 2. The solution has been constructed to permit spatidy-dependent velocity fields and

dispersivities. It is convenient to choose the following hnctions

Then

and

We suggest v~ = 0.23 1 and VT = 0.366. For i = 1,2, the velocity components are chosen to be:

o -,I,(r+y)lL vi = V, e ws(o it). (13)

Then

(no sum implied on right-hand-side). We suggest PI = 0.767, p2 = 0.536. Equation (8) then reduces to

- - again, with no sums implied. In summary, to evaluate the source term, use (6) with the auxiliary relations (5), (7), (2), (13), (14), (15). The initial condition is, of course,

Values must be supplied for the parameters UO, VO, a,, o2, 4, R, L, 1. and the m;udmum concentration, CM. Note that both the solution and the source are independent of the porosity and retardation parameters!

Discharge Calculation Let B be a subdomain of R with fi the unit outward normal for a point on the boundary 823 of the subdomain. The instantaneous mass discharge across dB is the line integral -

F = -DVc + vc

I fB is the rectangle [a, b] x [c, 4, then the line integral reduces to

n;r(t)=n;i,+n;r,+n;i,+MR

where

F, = -D1,c, - Dl2cY + UC,

Fy = -D12c, - D,cy + vc.

Formulas (Z), (3), (9), (lo), and (13) can be evaluated on the boundary to determine the 'analytic' instantaneous mass discharge (reducing the expression for the analytic discharge to quadratures).

2D Transport Benchmark Solutions - I1 P. Knupp Ecodynamics Research Associates, P.O. Box 9229, Albuquerque, NM 87 1 19

The following benchmark is designed to test the implementation of the implicit boundary conditions in the SECOTPZD transport code. The governing equation for this benchmark is:

where

This equation is obtained fiom the more general equation solved in the transport code by assuming

The elements of D are spatially independent, Free-water molecular di is ion is zero, Dual porosity term is off, Single-species decay. -

One way to construct a solution which has time-independent boundary conditions is to assume the concentration is of the form

: with CM a constant. This is a solution provided

Assume the domain is R = [0, L]' and let

ZJ(X,Y) = e -P(=+YYL

The choice u + v K=- L

CS.j id

gives g(x, y, t) = 0, which makes this test problem easy to implement. The initial condition is, of course,

- Values must be supplied for the nine parameters a& ar, u, v, 4, R, L, X, and the maximum concentration, c~ Note that both the solution and the source are independent of the porosity and retardation parameters! Various boundary conditions can be applied using (3) and (5). For example, to apply a Dirichlet condition to the face x = 0, the boundary condition would be

while the Neumann condition on that face would be

Pbne of Symmeby of Mabtx Slab

Detail for Fracture Spaang, b' Detail for Ftachrre Aperture, b, Clay Coating T h i i , bc, and Local Coordinate X

Figure 1 Schematic of dual-porosity model.

Figure 2 Schematic of Gte volume staggered mesh showing internal and ghost cells. The concentrations are deiined at cell centers and velocities at cell faces.

Fracture Concentration Time = 100 days

- Computation + Analytical. Tang

Figure 3 Fracture-Matrix coupling verification: comparison of computed hcture solution to the Tang analytical solution.

Matrix Concentration: Time = 100 days

- Computation. Z = 0.2875 m +Analytical. Tang - Computation. Z = 0.7625 m

+Analytical, Tang

Figure 4 Fracture-Matrix coupling verification: comparison of computed matrix solution to the Tang analytical solution.

0.9

+ 0.8 -

+ Fracture Concentration

0.7 - t sp1 - Computed. sp 1

-Analytical

0.6 - -

* 0.5 -

+

C 0.4 - * T

0.3 - - t

0.2 - i *

0.1 -

0.0 -

Figure 5 Multiple species verification: comparison of computed fracture solution to the Lester et. al. analytical solution for species 1.

Fracture Concentration - Computed, sp 2

- Analytical - Computed, sp 3

+Analytical

Figure 6 Multiple species veriiication: comparison of computed hcture solution to the Lester et. al. analytical solution for species 2 and 3.

Figure 7 Convergence test on PA problem, vector 2, temporal behavior of the source function.

Figure 8 Convergence test on PA problem, vector 2, hcture transport, concentrations contours and breakthrough curves for coarse grid 46x53.

Figure 9 Convergence test on PA problem, vector 2, fiacture transport, concentrations contours and breakthrough curves for medium grid 93x107. -

74

Figure 10 Convergence test on PA problem, vector 2, fracture transport, concentrations " ' . -.. -' contours and breakthough curves for h e grid 187x215.

, I I

80 120 180 ZOO i t 0 110 320 x m ~ t=1oa 1

Figure 1 1 Convergence test on PA problem, vector 52, dual-porosity transport, temporal behavior of the source function.

Figure 12 Convergence test on PA problem, vector 52, dual-porosity transport, concentrations contours and breakthrough curves for coarse grid 46x53.

-. Figure 14 Convergence test on PA problem, vector 52, dual-porosity transport, concentrations

contours and breaMhrough curves for tine grid 187x215.

79

Matrix Concentration TimeS00 days, Xa.9975 m - 5 uniform cells

10 uniform cells .-. 20 uniform cells - - . 40 uniform cells

80 uniform cells - Sudicky-Frind

Figure 15 Example problem: Sudicky-Frind, time = 500 days, comparison of grid convergence study to the analytical solution in the matrix.

Time = 500 days - Computed - Sudicky-Frind

1.0,

0.9 - + Fracture Concentration

0.8 - .

C 0 0.7 - n

0.6 - e n t 0.5 - r a 0.4 - t i o 0.3 -

n kg/rn3 0.2 -

0.1 -

0.0 -

Figure 16 Example problem: Sudicky-Frincl, time = 500 days, comparison of computed concentration profile to the analytical solution in the hcture.

Figure 17 Example problem: Sudicky-Frind, time = 500 days, comparison of computed concentration profile at different hcture locations to the analytical solutions in the matrix.

0.42

0.38

c 0.33 0 n C 0.28 e n t 0.23 r a t 0.18- I 0 n 0.13-

kg/m3 -

0.03 -\ - . , b ,, . , . . .

I . , , , , -

,9975 rn m " * .,. .?

-0.03 0.00 0.05 0.10 0.15 0.20 0.25

Distance into matrix. m

-i

+ Matrix Concentration -

r .so75 m Time = 500 days

- Computed (x =.3075 m) - * - Sudicky-Frind

i - Computed (x =.6975 m)

- Sudicky-Frind - i - Computed (x =.9975 m)

+ Sudicky-Frind +

- i - t

+ * t - ---.

Fracture Concentration + Steady State Solution

- Computed * Sudicky-Frind

f

+ .

+

+,.

rc, *.

X

x. x

'%.

\

. , I , , . , , , ,

F

Figure 18 Example problem: Sudicky-Frind, steady state solution, comparison of computed concentration profile to the analytical solution in the hcture.

- Figure 19 Schematic diagram of 2-D plane dispersion problem.

Figure 20

- ~~p -

Example problem: &actme transport for low Peclet case, analytical solution, Pe = 2.0.

Figure 21 Example problem: fiacture transport for low Peclet case, TVD limiter = 7; van Leer's MUSCL limiter, Pe = 2.0.

Figure 22 Example problem: hcture transport for low Peclet case, TVD limiter = 1; Upstream differencing, Pe = 2.0.

Figure 23 Example problem: hcture transport for high Peclet case, analytical solution, Pe = 10.0.

Figure 24 Example problem: hcture tmqort for high Peclet case, TVD limiter = 7; van Leer's MUSCL limiter, Pe = 10.0.

Figure 25 Example problem: liacture transport for high Peclet case, TVD limiter = 1; Upstream differencing, Pe = 10.0.

SECOTPXI. Vmion 1.30 WPO # 36695 User's Manual, Version 1.01 April 18. 1996

Pase 9

Appendix 11: Sample DiagnosticdDebug File

SSSSSS EEEEEEE CCCCC 00000 TTTTTT PPPPPP 2222 DDDDDD SS EE CC CC 00 00 TT PP PP 2 2 DD DD SS EE CC 00 00 TT PP PO 2 DD DD SSSSS EEEEE CC 00 00 TT PPPPPP 2 DD DD

SS EE CC 00 00 TT PP 2 DD DD SS EE CC CC 00 00 TT PP 2 DD DD

SSSSSS EEEEEEE CCCCC 00000 TT PP 222222 DDDDDD

SECOTP2D Version 1.2120 Version Date 08/16/93 Written by Kambiz Salari

Sponsored by rebecca blaine

Run on 09/22/95 at 14:15:32 Run on ALPHA AXP BEATLE OpenVMS V6.1

Prepared for Sandia National Laboratories Albuquerque, New Mexico 87185-5800 for the United States Department of Energy under Contract DE-AC04-76DP00789

* r * * * * * * x * * t * * * * * * * * * * * * * * * r t * * * * * * * * * * * ~ ~ * * * * * * *

Disclaimer ----------

This computer program was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government, any agency thereof or any of their contractors or subcontractors. The views and opinions

SECOTPZD, Version 1.30 User's Manual, Version 1.01

WPO # 36695 April 18. 1996

expressed herein do not necessarily state or reflect those of the United States Government, any agency thereof or any of their contractors or subcontractors.

FILE ASSIGNMENTS: -----------------

Number of chains = 1 Number of species = 1

Chain: 1 number of species I Species 1 = U233

Dual-porosity is used- Number of cells in the block, IMAX = 9

Number of cells in X-direction, JMAX = 46 Number of cells in V-direction, KMAX = 53

TIME BEGIN = 3.1557E+10 - TIME - END = 3.1557E+11

Time step is computed, DT = 5.6802~+08

Type of input velocity: time dependent

Maximum number of steps, NUMSTEP = 500

Algorithm parameters: Second order implicit 3-point backward time differencing is used. Coefficient of source term: AX = 0.50 AY = 0.50 Spatial differencing:

TVD, van Leer MUSCL limiter, LIMITER = 7

Property and velocity field data files: Property file = SECOTP.PRP

Velocity file = SECOTP.VEL

Intermediate results are printed at every 50 time steps.

SECOTPZD, Version 1.30 WPO # 36695 User's Manual, Version 1.01 April 18. 1996

Page 1 1 6

SECOTPZD CPU time i s 1 : 2 0 (minute:secondl

* * * END OF SECOTP2D * * * SECOTPZD 1 . 2 1 2 0 ( 0 8 / 1 6 / 9 3 ) 1 4 : 1 5 : 3 2

S E C O m D . Version 1.30 WPO # 36695 User's Manual. Version 1.01 April 18, 1996

Page 12

Appendix JII: Review Forms

This appendix contains the review forms for the SECOTP2D User's Manual.

NOTE: Copies of the User's Manual Reviewer's Foms are available in the Sandia WIPP Central Files.